cse5311-old/Assessment.htm010064400017600000145000000274001026440033200164330ustar00kumarfaculty00000400000024 Assessment
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Assessment and General Notes

Statement of Ethics

References

Notes and Exercises

Homeworks

Lab Assignments

Quizzes

Exams

Research Papers

Important Dates

Announcements

 

Assessment

Course grades will be based on the following:

Quizzes and class participation: 30%

 ACTIVE Participation will prepare you well for Quizzes and Exams. Students are expected to interact actively during lectures. All students are expected to solve homework problems and discuss solutions in the class.

There will be 2 quizzes.

The structure of the quizzes will be discussed in class, at least one week prior to the quiz.

 

Quiz 1 (15%) : September 14, 2005

Quiz 2 (15%) :  November 18, 2005

Exams: 35%

There will be ONE exam. The questions will test your ability to write and analyze algorithms. The exam will NOT test your memory. The exam will be a comprehensive open book/ notes, exam lasting for about 3-4 hours.

Exam:  October 31, 2005

Group Project/study : 35%

Students will have the option of doing a group study or group project.

Project problems will handed out at least one month prior to the expected date of completion. The students will be required to write programs and run experiments.

OR
 

Groups of students will be given a research problem to work on. Students are encouraged to come up with new and creative ideas.

A presentation of this work will  be on December 2, 2005.

Homework Assignments: No Grades awarded directly!

    

General Notes:

  •  The instructor reserves the right to modify course policies, the course calendar, and assignment    or project values and due dates

·   If you require any accommodation based on disability, please meet with the instructor  in the privacy of his office the first week of the semester to be sure you are appropriately accommodated

·      All students are expected to be responsible users of the computer systems for this course.

·      All students are expected to pursue their academic careers with honesty and integrity. Academic dishonesty includes, but is not limited to, cheating on a test or other course work, plagiarism (offering the work of another as one's own), and unauthorized collaboration with another person. Students found guilty of dishonesty in their academic pursuits are subject to penalties that may include suspension from the university.

·     Students are encouraged to discuss homework with classmates, but are not allowed to copy the solutions of others or share solutions with others. All work turned in for grading must be the student's own work.

·     A public access course website will be used as a repository for all course material. This directory will contain copies of any homework assignments, course handouts, project and paper lists, notes etc.

·    Students are expected to obtain accounts on any university computers needed for this class, and to be able to access the course repository and send and receive e-mail messages.

 PLEASE VISIT http://www2.uta.edu/discipline/ 

uizzes and Exams. Students are expected to interact actively during lectures. All students are expeccse5311-old/Assignments.doc010064400020320000200000001050001012440051200161050ustar00cse5311cse531100000400000024ࡱ> &(%5@ 0 bjbj22 (8XX9>>>>>>>R824f$R 'n(&&&&&&&z(R*&>000&>>&0>>&0& r;%T>>;& T %&&0 '%v+v+;&RR>>>>v+>;&XZ.@n4&&RRd DRR CSE 5311 Design and Analysis of Algorithms FALL 2004 Department of Computer Science The University of Texas at Arlington Assignment Problems Experimental Research Type 1. Write programs to implement sorting algorithms that employ different sorting algorithms, Quicksort, Mergesort, Selection sort, Heapsort etc. for sorting keys. Execute your sorting programs for the following sets of data: Set_1 Ordered List Set_2 - Reverse order List Set_3 A list containing the same value through out Set_4 Random List Set_5 25% of the List sorted PLEASE NOTE THAT your PROGRAMS WILL BE TESTED WITH DIFFERNET DATA SETS AS WELL Presentation of Results: Measure CPU time, memory space used and energy consumed for data sizes 1000, 10K, and 1M and more. Present your results using tables or graphs. Write a detailed analysis on the behavior of different algorithms Design a new sorting algorithm that consumes less energy than existing ones. Or identify an already published algorithm [for example see references] that will result in less energy consumption. You will be given a laptop (if you dont have one!) to implement your algorithms. All algorithms should be run when the laptop is on battery power only. Also you could do an analysis with programming languages e.g., Java vs C++. In each case, you have to interpret the results and justify the outcomes. The analysis can be as exhaustive as possible. References: [1] R.L. Wainwright, A Class of Sorting Algorithms based on Quicksort, Communications of the ACM, Vol. 28, No. 4, April 1985, pgs. 396-402. [2] R.S. Scowen, Algorithm 271: Quicksort, Communications of the ACM, Vol. 8, No. 11, Nov. 1965, pgs. 669-670. [3] M.N. vanEmden, Algorithm 402: Increasing the efficiency of Quicksort, Communications of the ACM, Vol. 13, No. 11, Nov. 1970, pgs. 693-694. [4] C.R. Cook, and Kim D.J, Best sorting algorithm for nearly sorted lists, Communications of the ACM, Vol. 23, No. 11, Nov. 1980, pgs. 620-624. Experimental Research Type 2. Consider a communication network that can be modeled as a weighted undirected connected graph G=(V,E). Each site in the network is represented as a vertex and each line of communication is bidirectional and has a cost associated with it. The cost may correspond to the expected delay on the line. Each vertex has only information about edges and vertices adjacent to it (i.e., local information). An MCST of the network can be used to broadcast messages to all sites. If we broadcast the messages by using only the edges of the MCST, then the global cost is minimized. Design an efficient algorithm to compute an MCST with any vertex v (V as the root. Assume now that the vertices in a certain subset U ( V share between them the information that is known to all of them. In other words, every vertex in U knows not only about edges and vertices adjacent to itself, but it also knows all the edges and vertices to all vertices in U. Also assume that the partial MCST restricted to U is connected (i.e., it is a tree). Now consider an edge e ( U which belongs to the MCST. Find conditions under which a change in e is guaranteed not to affect the MCST. Consider only conditions that can be checked with the information known to the vertex in U. In other words, how can the vertices in U determine that no action needs to be taken to modify the MCST after the change? Find the conditions under which the modified MCST is different from the original one only in edges that belong to U (hence the change can be handled locally). Consider only conditions that can be checked with the information known to the vertex in U. Describe an algorithm to check for the conditions in parts a and b (within U) and to modify the MCST accordingly. The algorithm does not need to handle the case where the change to the MCST may be outside U. In each case, implement your algorithm in code and execute your programs. Consider varying graph sizes ranging from 10s to 100s. Set_1 Connectivity of each vertex is 10 Set_1 Connectivity of each vertex is 4 Set_2 Connectivity is |V|/2 for each vertex Set_3 A Fully connected graph Programming and Presentation Problem 3 GREEDY STRATEGY and DYNAMIC PROGRAMMING You are required to submit the code, sample data, results, a report, and instructions for executing the code by email(mjoseph@cse.uta.edu) to the TA Presentation Problem: Perform an in-depth study of the following techniques: Greedy techniques and dynamic programming e.g., use knapsack and 0/1 knapsack problem for demonstration. The study must include, Presentation of the Algorithms Power point slides with animations Examples Analysis of the Algorithms Applications of the Algorithms and Working Program/Code for each of the above problems with sample graphs Completion of the Laboratory Assignment (described below) Please consult with the course instructor on a regular basis regarding the progress of your work and advice on extensions. You will receive credit for literature review and writing a comprehensive report on the problem. Laboratory Problem: Implement this problem for different data types (on real text) and show results. Consider the problem of printing a paragraph on a printer. The input text is a sequence of n words of length l1,l2,,ln, measured in characters. We want to print this paragraph neatly on a number of lines that hold a maximum of M characters each. Our criterion of neatness is as follows. If a given line contains words i through j and we leave exactly one space between words, the number of extra space characters at the end of the line is M-j+i- EMBED Equation.3 . We wish to minimize the sum, over all lines except the last, of the cubes of the numbers of extra space characters at the end of lines. Give a dynamic- programming algorithm to print a paragraph of n words neatly on a printer. Analyze the running time and space requirements of your algorithm. Identify one or more problems that have similar characteristics. Research Type Problem 4 Read the following paper, Goldberg A.V., Tarjan, A new approach to the maximum flow problem, Journal of the ACM, Vol. 35, October 1988, pgs. 921-940. Critical Analysis of the method. Can we improve on this algorithm? Show Analysis and experimentation required. 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cse5311-old/Exercise Set 3_spring.doc010064400020320000200000000570001011023274100176120ustar00cse5311cse531100000400000024ࡱ> (*)5@ v bjbj22 "XXv2222222F**** 6 F NNNNNNNN= ? ? ? ? ? ? R r? 2NNNNN? 22NNT NF2N2N= N=  Ve @22 NB `X*4 ! j 0 TPP FF2222P2 NNNNNNN? ? FF*FF*Exercise Set 3 Show that a depth-first search of an undirected graph G can be used to identify the connected components of G, and that the depth-first forest contains as many trees as G has connected components. Show how to modify depth-first search so that each vertex v is assigned an integer label cc[v] between 1 and k, where k is the number of connected components of G, such that cc[u] = cc[v] if and only if u and v are in the same connected component. Let G = (V,E) be an undirected graph such that each vertex has an even degree. Design a linear-time algorithm to direct the edges of G such that, for each vertex, the outdegree is equal to the indegree. Design an algorithm to find a vertex in a connected undirected graph whose removal does not disconnect the graph. The algorithm should run in linear time. Let G = (V,E) be an undirected weighted graph, and let T be an MCST of G. Suppose that we now add a new vertex v to G, together with some weighted edges from v to vertices of G. Design a linear-time algorithm to find a new MCST that includes v. Let (u, v) be a minimum-weight edge in a graph G. show that (u,v) belongs to some minimum spanning tree of G. Suppose that all edge weights in a graph are integers in the range from 1 to (V(. How fast can you make Kruskal's algorithm run? What if the edge weights are integers in the range from 1 to W for some constant W?  t u v h<`h<`5CJOJQJmH sH  jh<`CJOJQJmH sH h<`CJOJQJmH sH  9 : / s t u v  & Fgd<`gd<` $ & Fa$gd<`$a$gd<`v  1h/ =!"#$%@`@ <`NormalPJ_HmH nHsH tHDAD Default Paragraph FontRiR  Table Normal4 l4a (k@(No Listv9:/stux@0@0p@0@ 0@0@ 0@0@ 0@0@ 0@ 0@ 0@0@0@0v v v xlox/>mox333x x Mohan Kumar24@" ZFf<^`o(. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.hh^h`o(. hh^h`o(hH. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.4@"ZFf^x        t        <`tuxkr0@v`@UnknownGz Times New Roman5Symbol3& z Arial;|i0Batang"qh--  !24tt3QH)?<`Exercise Set 3 Mohan Kumar Mohan Kumar   Oh+'0x  4 @ LX`hpExercise Set 3xer Mohan KumarohaohaNormalu Mohan Kumar1haMicrosoft Word 10.0@G@kkX@xX՜.+,0 hp  CSE@UTA tA Exercise Set 3 Title  !"%Root Entry FX'1Table WordDocument"SummaryInformation(DocumentSummaryInformation8CompObjj  FMicrosoft Word Document MSWordDocWord.Document.89qRoot Entry F [X-@1Table WordDocument"SummaryInformation( ,+ _AdHocReviewCycleID_EmailSubject _AuthorEmail_AuthorEmailDisplayName 4sExercise Set 3kumar@cse.uta.edu Mohan KumarDocumentSummaryInformation8CompObjj  FMicrosoft Word Document MSWordDocWord.Document.89q՜.+,D՜.+,@ hp  CSE@UTA tA Exercise Set 3 Title@ 4`cse5311-old/Exercise Set 4.doc010064400020320000200000000560001011023274200162310ustar00cse5311cse531100000400000024ࡱ> ')(5@  bjbj22 "XX6666666J  J=??????$R rc6c66xTTT 66=T=T TtV@66 PF =00t "(t JJ6666t 6 TccJJdJ JJExercise Set 4 Let G = (V,E) be a connected undirected weighted graph. Assume for simplicity that weights are positive and distinct. Let e be an edge of G. Denote by T(e) the spanning tree of G that has minimum cost among all spanning trees of G that contain e. Design an algorithm to find T (e) for all edges e ( E. The algorithm should run in time O((V(2). Give an algorithm that determines whether or not a given undirected graph G = (V,E) contains a cycle. Your algorithm should run in O((V() time. Let G=(V,E) be a directed weighted graph such that all the weights are positive. Let v and w be two vertices in G and k ( (V( be an integer. Design an algorithm to find the shortest path from v to w that contains exactly k edges. The edge path need not be simple.   # $ & ' 7 8 9 : ; < _ ` a b c d e j          򺯺h7tOJQJ jhtOJQJhthtCJOJQJ jhtOJQJhtOJQJhtCJH*OJQJ jhtCJOJQJ jhtCJOJQJht6CJOJQJhtCJOJQJht5i j        $h^ha$gd7t$a$gd7t$a$gdt $h^ha$gdt $ & Fa$gdt   1h/ =!"#$%D@D NormalCJ_HaJmH nHsH tHDAD Default Paragraph FontRiR  Table Normal4 l4a (k(No List  ij 00 00 00 000p00    _a 33333   Mohan Kumar0?E al &v hh^h`o(.^`o(. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.hh^h`o(.0?Eal&v        7tt@|+@UnknownGz Times New Roman5Symbol3& z Arial;|i0Batang"hȅ&ȅ&ww243QH)?tExercise Set 4 Mohan Kumar Mohan Kumar   Oh+'0x  4 @ LX`hpExercise Set 4xer Mohan KumarohaohaNormalu Mohan Kumar2haMicrosoft Word 10.0@@xL@xLw՜.+,0 hp  CSE@UTAA Exercise Set 4 Title  !$Root Entry F &1Table WordDocument"SummaryInformation(DocumentSummaryInformation8CompObjj  FMicrosoft Word Document MSWordDocWord.Document.89qRoot Entry F",@1Table WordDocument"SummaryInformation( +* _AdHocReviewCycleID_EmailSubject _AuthorEmail_AuthorEmailDisplayName DExercise SET 4kumar@cse.uta.edu Mohan KumarDocumentSummaryInformation8CompObjj  FMicrosoft Word Document MSWordDocWord.Document.89q՜.+,D՜.+,@ hp  CSE@UTAA Exercise Set 4 Title@ 4`cse5311-old/Exercise Set II.doc010064400020320000200000000500001011023274200163610ustar00cse5311cse531100000400000024ࡱ> #%"7 ) bjbjUU "7|7|l  G$ -XXXX^XXbX:n,  : 0G XXExercise Set II Show that the n element heap has height (log n( Show that the largest element in a subtree of a heap is at the root of the subtree. Show that the worstcase running time of HEAPIFY on a heap of size n is ((n). Show that quicksorts best case running time is ((n logn) Suppose that the splits at every level of quicksort are in the proportion 1-a to a, where 0 < a ( is a constant. Show that the minimum depth of a leaf in the recursion tree is approximately (logn/log a) and the maximum depth is approximately (log n / log (1-a)) (don t worry about integer round-off). The input is a set S containing n real numbers, and a real number x. Design an algorithm to determine whether there are two elements of S whose sum is exactly x. The algorithm should run in O(n log n) time. Suppose now that the set S is given in a sorted order. Design an algorithm to solve the above problem in time O(n). 89>?  $&(*tv( )  jOJQJOJQJ jWOJQJ jOJQJ jOJQJOJQJ5CJOJQJ@*( )  $ & Fdha$ $ & Fdha$ & Fdhdh^ )  1h/ =!"#$% i8@8 NormalCJ_HaJmH sH tH <A@< Default Paragraph Font@L0 0 0 0 0 0 0 0 0 0) ) ) cjEN}  3333Savita ChauhanC:\cse5311\Exercise Set II.docSlHTdXG Vrl688^8`o(.^`o(.SlG Vr@mY(ggP@PP@UnknownGz Times New Roman5Symbol3& z Arial"qhc 4655@  bjbj22 "XXF<vvvvb b b b ~  uuuRD!tu)Luuuvv uv  u N Cb uR0!!vvvv! uuuuuuu  Exercise Set VI 1. Prove that for any pair of vertices u and v and any capacity and flow function c and f, we have r(u,v) + r(v,u) = c(u,v) + c(v,u) 2. Determine the maximum flow for the given network            3. Given a flow network G = (V, E), let f1 and f2 be functions from V ( V to R. The flow sum f1 + f2 is the function from V ( V to R defined by (f1 + f2)(u, v) = f1(u, v) + f2(u, v) for all u, v ( V. If f1 and f2 are flows in G, which of the three flow properties must the flow f1 + f2 satisfy, and which might it violate? 8 V2 V1 12 10 6 8 s s 2 8 7 6 16 V4 V3 12       % & ' , - . 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In each case, say whether f(n) = O(g(n), f(n) = ( (g(n)), and/or f(n) = ( (g(n)) f(n) g(n) a. 100n +log n n + (log n)2 b. log n log(n2) c. n2/log n n(log n)2 d. (log n)log n n/log n e. (n (log n)5 f. n 2n 3n Suppose that T(n) is O(f(n)) and g(n) is a function whose value is never negative. Prove that g(n)T(n) is O(g(n)f(n)). Exercise Set II Show that the n element heap has height (log n( Show that the largest element in a subtree of a heap is at the root of the subtree. Show that the worstcase running time of HEAPIFY on a heap of size n is ((n). Show that quicksorts best case running time is ((n logn) Suppose that the splits at every level of quicksort are in the proportion 1-a to a, where 0 < a ( is a constant. Show that the minimum depth of a leaf in the recursion tree is approximately (logn/log a) and the maximum depth is approximately (log n / log (1-a)) (don t worry about integer round-off). The input is a set S containing n real numbers, and a real number x. Design an algorithm to determine whether there are two elements of S whose sum is exactly x. The algorithm should run in O(n log n) time. Suppose now that the set S is given in a sorted order. Design an algorithm to solve the above problem in time O(n). %+,:Jnr{ҶҪ҅yyyn`n jCJOJQJmH sH CJOJQJmH sH CJH*OJQJmH sH  jCJOJQJmH sH CJH*OJQJmHsHCJOJQJmHsH5CJOJQJmHsH jQCJOJQJmH sH  jWCJOJQJmH sH CJOJQJmH sH 5OJQJmH sH 5CJOJQJmH sH 5CJOJQJmH sH !%Nm45JpUy{ & Fdh hdh^`hdhpdh^p8dh^8 & Fdh & Fdhdh^) VW  $&(*  t v ( ) mH sH  jCJOJQJmH sH CJOJQJmH sH  jWCJOJQJmH sH CJOJQJmH sH  jCJOJQJmH sH \ * ( )  $ & Fdha$ $ & Fdha$ & Fdh 1h/ =!"#$% i8@8 NormalPJ_HmH nHsH tH<A@< Default Paragraph Font %Nm45JpUy{\   0000 0 0 0 000000000 0000 0 0 0 0 0 0 0 0)  )  ) :JLpr &fqXY] NPmo:ILor`b46 33333333333%{  Mohan KumarSavita Chauhan C:\cse5311\Exercise SetsI&II.docSavita ChauhanPC:\Documents and Settings\Administrator\Desktop\Cse5311Web\Exercise SetsI&II.doc@ ~»q_SlHTdXG Vrl688^8`o(.^`o(.88^8`o(.^`o(.SlG Vr@ q_ !0@33ܖt33( @@@@@UnknownGz Times New Roman5Symbol3& z Arial; Batang"qh'܁F'܁F % !20 3QHExercise Set I Mohan KumarSavita ChauhanOh+'0   < H T`hpxExercise Set Ixer Mohan Kumarohaoha Normal.dotSavita Chauhan2viMicrosoft Word 9.0@@P[&@P[&՜.+,D՜.+,@ hp  CSE@UTA  Exercise Set I Title8_AdHocReviewCycleID_EmailSubject _AuthorEmail_AuthorEmailDisplayNameExercise -1 cse5311kumar@cse.uta.edu1 Mohan Kumar  !"#$'Root Entry Fp9i&)1Table WordDocument"SummaryInformation(DocumentSummaryInformation8CompObjjObjectPoolp9i&p9i&  FMicrosoft Word Document MSWordDocWord.Document.89qcse5311-old/Exercise1.doc010064400020320000200000000650001011255622100154550ustar00cse5311cse531100000400000024ࡱ> 02/5@  #bjbj22 "(XXxxxxxxx  <  ACCCCCC$RHVg9x gxx ., , , ^x x A, A, ,, X V@xx P Z  A< 0 .xxxxx@ , gg$4d" 4Exercise Set I CSE 5311 Design and Analysis of Algorithms Fall 2004 Solve the following recurrence relations T(n) = T(n-1) + n/2 , T(1) = 1 T(n) = 8T(n-1) -15T(n-2), T(1) = 1, T(2) = 4 Give asymptotic upper and lower bounds for T(n) in each of the following recurrences. Assume that T(n) is constant for n d" 2. Make your bounds as tight as possible, and justify your answers. T (n) = 2T(n/2) + n3 T(n) = T(9n/10) + n T(n) = 16T(n/4) + n2 T(n) = 3T(n/2) + n lg n T(n) = T(n-1) + lg n Argue that the solution to the recurrence T(n) = T(n/3) + T(2n/3) + n is &!(n lg n) by appealing to a recursion tree. Use a recursion tree to solve the recurrence T(n) = T(n) + T((1-)n) + n, Where  is a constant in the range 0<  < 1. 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In each case, say whether f(n) = O(g(n), f(n) = ( (g(n)), and/or f(n) = ( (g(n)) f(n) g(n) a. 100n +log n n + (log n)2 b. log n log(n2) c. n2/log n n(log n)2 d. (log n)log n n/log n e. (n (log n)5 f. n 2n 3n H ~!!!,"p"""## #dhgda! pdh^pgda!L & FdhEƀP&.gda! !!!&"("8":"l"t""""""""""##### #յյյ՗h^Wh^WCJOJQJmH sH hl9PCJOJQJaJmH sH ha!CJH*OJQJmH sH  jha!CJOJQJmH sH ha!CJOJQJmH sH ha!CJH*OJQJmHsHha!CJOJQJmHsH 1h/ =!"#$%<@< ^WNormal_HmH nHsH tHDAD Default Paragraph FontRiR  Table Normal4 l4a (k(No ListHH LiA Balloon TextCJOJQJ^JaJ(Enz@0@0@ 0@ 0@ 0@0@ 0@ 0@ 0@ 0@ 0Oy00XEnz/[\:WX.Jl000 0 0 0@0 0 0 0 0 0 0 0@0@0 0@0@0 0@0@0@0 0@0@0@0 @0 @0@0 @00@0Enzl@0@0@0@ 0@ 0@ 0@0@ 0@ 0@ 0@ 0@ 0@H0My00PsMy00My00My00@0My00 ! #n  P H #  #  .0HLQRln}/1DF -0ILkn3333333nz Pico Mohan Kumar@ ~»q_![t R88^8`o(.^`o(.^`o(. pp^p`hH. @ L@ ^@ `LhH. ^`hH. ^`hH. L^`LhH. ^`hH. PP^P`hH.  L ^ `LhH.@ q_![t          "a!LiA PNl9P=c8^WP>x@ 4444@@@ @ @ Unknown Mohan KumarGz Times New Roman5Symbol3& z Arial5& zaTahoma"hS&BF !24d 3QH)?^WExercise Set IPicoPico   Oh+'0p  , 8 DPX`hExercise Set IxerPicoiseicoicoNormalePicole5coMicrosoft Word 10.0@e@v@ޟ՜.+,0 hp  CSE@UTAA Exercise Set I Title  !"#$%&()*+,-.1Root Entry Fpl31TableWordDocument"(SummaryInformation(DocumentSummaryInformation8'CompObjj  FMicrosoft Word Document MSWordDocWord.Document.89qcse5311-old/Exercise2.doc010064400020320000200000000760001011255624000154610ustar00cse5311cse531100000400000024ࡱ> 9;85@ Y'bjbj22 ",XX|||||||\ \ \ \ p Dx $dR\Q| || 8 j| |   , %||% `矉\ * v]$0Hx\ 4 (%|||| |%8 ddX XExercise Set II CSE 5311 Design and Analysis of Algorithms Fall 2004 What are the minimum and maximum numbers of elements in a heap of height h? Show that an n-element heap has a height (log n(. Show that the largest element in a subtree of a heap is at the root of the subtree. Show that the worst case running time of HEAPIFY on a heap of size n is &!(lg n). Illustrate the operation of Heap-Extract-Max on a heap A = {15,13,9,5,12,8,7,0,6,2,1} 6) A d-ary heap is like a binary heap, but instead of 2 children, nodes have d children. How would you represent a d-ary heap in an array? What is the height of a d-ary heap of n elements in terms of n and d. Give an efficient implementation of EXTRACT-MAX. Analyze its running time in terms of d and n. Give an efficient implementation of INSERT. Analyze its running time in terms of d and n. If the input is d sequences of elements such that each sequence is already sorted, and there is a total of n elements. Design an O(n log d) algorithm to merge all the sequences into one sorted sequence. 7) Suppose that the splits at every level of quicksort are in the proportion 1-  to , where 0 <  ( is a constant. Show that the minimum depth of a leaf in the recursion tree is approximately (logn/log  ) and the maximum depth is approximately (log n / log (1- )) (don t worry about integer round-off). 8GH  ` b 0 2 6 7   a c w Ͼtbbbbbbbbbbbb#hodqhkP6CJOJQJ^JaJ hodqhr`8CJOJQJ^JaJ hodqhkPCJOJQJ^JaJ& jhodqhrCJOJQJ^JaJ& jhodqhrCJOJQJ^JaJ hodqhrCJOJQJ^JaJhodqh*5OJQJ^Jh*5OJQJhr5OJQJ^Jhodqhr5OJQJ^J&=GHI hdh^hgdr`8L & FdhEƀt&)gdr`8dhgdr`8$dh^a$gd*@ dh^@ `gdr`8Y'  ]L & FdhEƀt&)gdr`8dhgdr`8L & FdhEƀt&)gdr`8 i   MA hdh^hgdr`8L & FdhEƀt&)gdr`8dhgdr`8hhdh^h`hgdr`8L & FdhEƀt&)gdr`8 ~ ; PO$ & FdhEƀt&.a$gdr`8O$ & FdhEƀt&.a$gdr`8$hdh^ha$gdr`8; _O$ & FdhEƀt&.a$gdr`8O$ & FdhEƀt&.a$gdr`8w x y XZ`$$z$|$~$$&&N'Q'Y'Ⱥܦt`tttUJhr`8hkPCJaJhr`8hr`8CJaJ'Hh*FhE@CJOJQJ^JaJ hodqhr`8CJOJQJ^JaJUhE@CJOJQJ^JaJ hodqhE@CJOJQJ^JaJ'Hh*FhkPCJOJQJ^JaJh|}*CJOJQJ^JaJ& jhodqhkPCJOJQJ^JaJ hodqhkPCJOJQJ^JaJ#hodqhkP6CJOJQJ^JaJ z$|$&CCQ$YdhC$Eƀ*F^Ya$gdr`8$Ydh^Ya$gdr`8 $dha$gdr`8O$ & FdhEƀt&.a$gdr`8) Show that quicksort s best-case running time is &!(n lg n). 9) The input is d sequences of elements such that each sequence is already sorted, and there is a total of n elements. Design an O(n log d) algorithm to merge all sequences into one sorted sequence. 10). The input is a set S containing n real numbers, and a real number x. Design an algorithm to determine whether there are two elements of S whose sum is exactly x. The algorithm should run in O(n log n) time. Suppose now that the set S is given in a sorted order. Design an algorithm to solve the above problem in time O(n). &&P&&N'O'P'Q'PHdhgdr`8O$ & FdhEƀt&.a$gdr`8O$ & FdhEƀt&.a$gdr`8$Ydh^Ya$gdr`8Q'R'S'T'U'V'W'X'Y' hdh^hgdr`8$dh^a$gdr`8 $dha$gdr`8 1h/ =!"#$%@@@ NormalCJ_HaJmH sH tH DAD Default Paragraph FontRiR  Table Normal4 l4a (k(No ListH@H * Balloon TextCJOJQJ^JaJ,=G@0@0 0<3=GHIirs(\_+,ghnoC00000 00 00 00 000 000 0 0 0 0 000000000 0 000000000000=GHIirs(\_+,ghn00000 00 00 00 000 000 0 0 0 0 0M900M900M90 0My0 0@0@0@0A 0@00iw Y' ; &Q'Y' Y' EHvyhjuikh(*tw2433333333=IirotPico Mohan KumarH\ .88\ (d\ ;g\ SlHTdXG Vrl6L wʪ^`o() ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. 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L LLhV H..n  08_Zf: 0  T4n BB.....n 0L BB$f6BBfExercise Set 3 Show that a depth-first search of an undirected graph G can be used to identify the connected components of G, and that the depth-first forest contains as many trees as G has connected components. Show how to modify depth-first search so that each vertex v is assigned an integer label cc[v] between 1 and k, where k is the number of connected components of G, such that cc[u] = cc[v] if and only if u and v are in the same connected component. Let G = (V, E) be an undirected graph such that each vertex has an even degree. Design a linear-time algorithm to direct the edges of G such that, for each vertex, the outdegree is equal to the indegree. Design an algorithm to find a vertex in a connected undirected graph whose removal does not disconnect the graph. The algorithm should run in linear time. Let G = (V, E) be an undirected weighted graph, and let T be an MCST of G. Suppose that we now add a new vertex v to G, together with some weighted edges from v to vertices of G. Design a linear-time algorithm to find a new MCST that includes v. Let (u, v) be a minimum-weight edge in a graph G. show that (u, v) belongs to some minimum spanning tree of G. Suppose that all edge weights in a graph are integers in the range from 1 to (V(. How fast can you make Kruskal's algorithm run? What if the edge weights are integers in the range from 1 to W for some constant W? 7. Let G = (V, E) be a connected undirected weighted graph. Assume for simplicity that weights are positive and distinct. Let e be an edge of G. Denote by T(e) the spanning tree of G that has minimum cost among all spanning trees of G that contain e. Design an algorithm to find T (e) for all edges e ( E. The algorithm should run in time O((V(2). 8. Give an algorithm that determines whether or not a given undirected graph G = (V, E) contains a cycle. Your algorithm should run in O((V() time. 9. Let G= (V, E) be a directed weighted graph such that all the weights are positive. Let v and w be two vertices in G and k ( (V( be an integer. Design an algorithm to find the shortest path from v to w that contains exactly k edges. The edge path need not be simple. 10. Let G = (V, E) be a directed acyclic graph, and let k be the maximal number of edges in a path of G. Design an algorithm to divide the vertices into at most k+1 groups such that for each two vertices v and w in the same group there is no path from v to w and there is no path from w to v. The algorithm should run in linear time. 11. Let G = (V, E) be a directed graph and let v and w be two vertices in G. Design a linear time algorithm to find the number of different shortest paths (not necessarily vertex disjoint) between v and w. (There are no weights on the edges). 12. Given a flow network G = (V, E), let f1 and f2 be functions from V ( V to R. The flow sum f1 + f2 is the function from V ( V to R defined by (f1 + f2)(u, v) = f1(u, v) + f2(u, v) for all u, v ( V. 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PLP^P`LhH.ZFfal4@"{PPt                MK~b0uK+@ .. 5 @@UnknownGz Times New Roman5Symbol3& z Arial"h3&*4&c c 24/ / 3QH)?KExercise Set 3 Suchitha Suchitha   Oh+'0|  8 D P\dltExercise Set 3xer SuchithaetSucSuc Normal.dott Suchithat3ucMicrosoft Word 10.0@캃@]I@VZc ՜.+,0 hp|  UTA/ A Exercise Set 3 Title  !"#$&'()*+,/Root Entry FP_Z11TableWordDocument"&SummaryInformation(DocumentSummaryInformation8%CompObjj  FMicrosoft Word Document MSWordDocWord.Document.89qcse5311-old/HA1FALL04.doc010064400020320000200000001120001012265630700150010ustar00cse5311cse531100000400000024ࡱ> EGD5@ 0bjbj22 (,XX; 86,b$$n($$$$$$$$%RQ(6$GGG6$K$G$G$ `"# p e,##a$0$F#b((4#(#LZ@T46$6$ $ CSE 5311 Design and Analysis of Algorithms FALL 2004 Department of Computer Science The University of Texas at Arlington Home Assignment I Quiz 1 (09/23/04) will be based on these questions Answer questions 1a to 1c pertaining to the STOOGE_SORT Algorithm given below. STOOGE_SORT(A,i,j) if A[i] > A[j] then exchange A[i] ( A[j] if i+1 ( j then return k ( ((j-i+1)/3( STOOGE_SORT(A,i,j-k) STOOGE_SORT(A,i+k,j) STOOGE_SORT(A,i,j-k) Argue that STOOGE_SORT (A,1,length[A]) correctly sorts the input array A[1 .. n], where n = length[A]. Give a recurrence for the worst-case running time of STOOGE_SORT and a tight asymptotic bound on the worst-case running time. Compare the worst-case running time of STOOGE_SORT with that of insertion sort, merge sort, heapsort, and quicksort. Given an array of integers A[1..n], such that, for all i, 1 ( i < n, we have (A[i]-A[i+1]( ( 1. Let A[1] = x and A[n] = y, such that x < y. Design an efficient search algorithm to find j such that A[j] = z for a given value z, x ( z ( y. What is the maximal number of comparisons to z that your algorithm makes? What is Counting Sort? Give the Algorithm and provide analysis. What are Fibonacci Heaps? Give an example of a Fibonacci Heap. Write basic algorithms for insertion and deletion in Fibonacci Heaps. You are given a collection of n bolts of different widths and n corresponding nuts. You are allowed to try a nut and bolt together, from which you can determine whether the nut is larger than the bolt, smaller than the bolt, or matches the bolt exactly. However there is no way to compare two nuts together or two bolts together. The problem is to match each bolt to its nut. Design an algorithm for this problem with average case efficiency of ((n log n). Explain how we can check a graphs acyclicity by using Bredth-first search. Does either of the two traversals DFS or BFS always find a cycle faster than the other? If your answer is yes, indicate which of them is better and explain why it is the case; if you answer no, give two examples supporting your answer. A graph is said to be bipartite if all its vertices can be partitioned into two disjoint subsets X and Y such that every edge connects a vertex in X with a vertex in Y. a. Design an DFS-based algorithm for checking whether a graph is bipartite. b. Design a BFS-based algorithm for checking whether a graph is bipartite. Prove that the number of different paths of length k>0 from the ith vertex to the jth vertex of a graph(undirected or directed) equals the (i,j)th element of Ak, where A is the adjacency matrix of the graph. [ to be discussed in class]. You are given a list of numbers for which you need to construct a min-heap. (A Min-heap is a complete binary tree in which every key is less than or equal to the keys in its children.) Write a min-heap algorithm and analyze its complexity. Suppose we are to find the k smallest elements in a list of n elements, and we are not interested in their relative order. Can a linear-time algorithm algorithm be found when k is a constant? If so give the algorithm. In either case, justify your answer. Modify Dijkstras single source shortest path algorithm so that it checks if a directed graph has a cycle. Analyze your algorithm. Design a divide-and conquer algorithm for polynomial evaluation. How many additions and multiplications does your algorithm require? Compare your algorithm with that based on Horners rule in terms of efficiency. GUIDELINEs for 1-pg Reference Sheet Printed on only one side of the Sheet One inch (or 2.5 cms) Margin on all four sides Single Column format Font Type: Times Roman or Times New Roman or Aerial Minimum Font Size : 11 Your full name should be written on the sheet CSE 5311 Fall 2004 Page  PAGE 1 of  NUMPAGES 2 +26z{ ! 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PLP^P`LhH. hh^h`o(hH) ^`hH) 88^8`hH) ^`hH() ^`hH() pp^p`hH()   ^ `hH. @ @ ^@ `hH.   ^ `hH.^`o() ;3|fk6dT$klIx0?EQ@=6s~zMx9' T        hD          Gg_dt+e0CH: ^ 9'x5r-@@UnknownG: Times New Roman5Symbol3& : Arial;|i0Batang?5 z Courier New;Wingdings"qhJ83 3 243QH)?e0*CSE 5311 Design and Analysis of Algorithms Mohan Kumar Mohan Kumar4         Oh+'0 (4 P \ ht|+CSE 5311 Design and Analysis of AlgorithmsoSE  Mohan KumarohaohaNormalu Mohan Kumar8haMicrosoft Word 10.0@&>|@x@؟3 ՜.+,0 hp  CSE@UTAA +CSE 5311 Design and Analysis of Algorithms Title  !"#$%&'()*+,-./012356789:;=>?@ABCFRoot Entry F@-HData 1Table/)WordDocument(,SummaryInformation(4DocumentSummaryInformation8<CompObjj  FMicrosoft Word Document MSWordDocWord.Document.89qcse5311-old/HASP042.doc010064400017600000145000000570001004153742200152470ustar00kumarfaculty00000400000024ࡱ> 5@ 0)$bjbj22 "&XX) ~~~~~~~,& F\\\\\\\0 2 2 2 2 2 2 RX2 -~T \\T T 2 ~~\\_  T ~\~\0 T 0 : ,~~ \: pq6 v  u 0 A "A ~~~~A~ 0\hJ <J \\\2 2  CSE 5311 Design and Analysis of Algorithms SPRING 2004 Department of Computer Science The University of Texas at Arlington Home Assignment 2 Quiz on Home Assignment: April 28, 2004, 10:30 AM in Room 308 WH The largest common subsequence (LCS) of two sequences T and P is the largest sequence L such that L is a subsequence of both T and P. The smallest common supersequence (SCS) of two sequences T and P is the smallest sequence L such that both T and P are subsequences T and P are subsequences of L. Design efficient algorithms to find the LCS and SCS of two given sequences. Let d(T,P) be the smallest edit distance between T and P such that no replacements are allowed (in other words, we have to insert or delete). Prove that d(T,P) = (SCS(T,P)(- (LCS(T,P)(, where (SCS(T,P)(is the size of the smallest SCS of T and P and (LCS(T,P)(is the smallest LCS of T and P. The input is two strings of characters A = a1 a2 ( an and B = b1 b2 ( bn . Design an O(n) algorithm to determine whether B is a cyclic shift of A. In other words, the algorithm should determine if there exists an index k, 1 ( k ( n such that ai = b(k+i) mod n, for all i, 1 ( i ( n. Suppose that each source si in a multisource, multisink problem produces exactly pi units of flow, so that f(si,V) = pi. Suppose that each sink tj consumes exactly qj units so that f(V,tj) = qj, where  EMBED Equation.3 . Show how to convert the problem of finding a flow f that obeys these additional constraints into the problem of finding a maximum flow in a single-source, single-sink flow network. Let G be a connected, undirected graph with at least 3 vertices, and let G3 be the graph obtained by connecting all pairs that are connected by a path in G of length at most 3. Prove that G3 is Hamiltonian. The subgraph isomorphism problem takes two graphs G1 and G2 and asks whether G1 is isomorphic to a subgraph of G2. Show that the subgraph isomorphism problem is NP-complete. Given an integer mn matrix and an integer m-vector b, the 0-1 inter-programming problem asks whether there is an integer n-vector x with elements in the set {0,1} such that Ax d" b. Prove that 0-1 integer programming is NP-complete. The input is a set of n rectangles all whose edges are parallel to the axes. Design an algorithm to fin+-8|}   ( ) 5 6 P Q V W M N O P Q R { | ͽ jhICJOJQJhI6CJOJQJhICJOJQJh9GCJOJQJhe0h9G5CJOJQJaJh9G5CJOJQJaJh9GCJOJQJaJhe0h9GCJOJQJaJhe0h9GCJaJ9+-9X} I n o d ()$a$gdH$a$gdI $h`ha$gdI $h^ha$gdI $ & Fa$gdI $ & Fa$gdI$a$gd9G$a$gd9Ggd9Ggd9G)$          0 3 7 8 = > C D G H I J K L M ] ` d e j k  K L  jhICJOJQJhI6CJH*OJQJhI6CJOJQJ jhICJOJQJhICJOJQJML P Q R S T U V W d e f i j k l m n p u      " 3 4 5 D E F G H I J N O P X Y l jhICJOJQJUhICJH*OJQJhI6CJH*OJQJhI6CJOJQJ jhICJOJQJhICJOJQJ@l m n o ()-.rst-./456HIJjklzzzzp`huO1huO16CJOJQJ]huO1CJOJQJ!hIhI6CJH*OJQJ]!hIhI6CJH*OJQJ]hIhI6CJOJQJ]hHCJOJQJhI6CJOJQJhICJOJQJjhICJOJQJUjhICJEHOJQJU,js< hICJOJQJUVmHnHu#.02&$'$)$$a$gduO1 $ & Fa$gdI$a$gdH JL\`,.2$'$($)$ԾԺhuO1hICJOJQJhIUhHCJOJQJhICJOJQJhuO1CJOJQJhuO1huO16CJOJQJ]"huO1huO16CJOJQJ]^Jd the intersection of all rectangles. 1h/ =!"#$%LDd B  S A? 2/&{D`!/&{՜` @CXxcdd``cd``baV d,FYzP1C&,7\N! KA?HZ @+깡jx|K2B* R&]_c&,`Bܤ\jޖ A|.B]j m1i%oCoB.dA|`ܞ, #l* eZZǀnOdwCeȤ2! ~|G`2>?Y|[ vGa%_J.hqclnv0o8M021)W2dO (c3X?w !"-#%&'()*+,Root Entry  F@! Data WordDocument "&ObjectPool pq@!_1015054707FpqpqOle CompObjfObjInfo  FMicrosoft Equation 3.0 DS Equation Equation.39q_V p ii " =q jj "Oh+'0 Equation Native r1Table$USummaryInformation( DocumentSummaryInformation8  0 < H T`hpx1ss Mohan KumarohaohaNormalu Mohan Kumar1haMicrosoft Word 10.0@@6:@勇@zԎ^՜.+,0 hp  CSE@UTA% A 1 Title  FMicrosoft Word Document MSWordDocWord.Document.89q@`@ INormalPJ_HmH nHsH tHDA@D Default Paragraph FontRi@R  Table Normal4 l4a (k@(No ListJJ`J 9GSubtitle$a$5CJ(OJQJnHtH) &+-9X}Inod()& ' + @0@0@0@0@0@0@0@0@ 0@ 0@ 0@0@ 0@0@0@ 0@0@ 0@0@0@ 0@0@ 0@0@0@ 0@0@0+-9X}+ @0@0@0@0@0@0 0!H L l )$ )$ )$ Xln) :5M5M5M\a5Mw#5Mf5M5M\\jss+ foo||+ =*urn:schemas-microsoft-com:office:smarttags PlaceType=*urn:schemas-microsoft-com:office:smarttags PlaceName9*urn:schemas-microsoft-com:office:smarttagsplace8*urn:schemas-microsoft-com:office:smarttagsCity8*urn:schemas-microsoft-com:office:smarttagsdate8*urn:schemas-microsoft-com:office:smarttagstime  10200428304DayHourMinuteMonthYear mzdfkn!35FJNP^f|+ MOdfDF+ 333333333' ( + +  Mohan Kumar;3  NlRchh^h`o(.^`o(.;3 NHuO1;C:aI9G}+ D@( ) `@``H@UnknownGz Times New Roman5Symbol3& z Arial;SimSun[SO"qhbwƅ^^!24% % 3Q H)?;C1 Mohan Kumar Mohan Kumar  CompObjjcse5311-old/Homeworks.htm010064400020320000200000000044641012265635300156450ustar00cse5311cse531100000400000024 Homeworks
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cse5311-old/ImportantDates.htm010064400017600000145000000051431026440121100172410ustar00kumarfaculty00000400000024 Important Dates
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                                                                  Important Dates

  • Classes begin  - August 22, 2005

  • Quiz 1 September 14

  • Quiz 2 November  18

  • Exam October 31

  • Group Work Presentation December 2


 

cse5311-old/LAB1.doc010064400020320000200000000650001012437723200143120ustar00cse5311cse531100000400000024ࡱ> 02/5@ _bjbj22 "XX_6666666J....JJ jjjjjjjj       $ R 6 6!jj!!6 66jjK !:6j6j ! 0l 66 j^ P@.["  a 0 ,u}.u JJ6666u6 Xj~ jjj6 6 JJ.JJ.CSE 5311 Design and Analysis of Algorithms Laboratory Assignment I The Assignment must be turned in by November 1, 2004 You are required to submit the code, sample data, results, a one page report, and instructions for executing the code by email(mjoseph@cse.uta.edu) to the TA Assignment Problem: The Floyd-Warshalls algorithm for all pairs shortest paths problem is applicable to Graphs in general. However, this algorithm involves a great deal of unnecessary work if the graphs are sparse. The Johnsons algorithm (see Chapter 25 in the Text Book) is more efficient for sparse graphs. Compare the performance of the Floyd-Warshalls and Johnsons algorithms for sparse graphs. Write programs to implement your algorithms Floyd-Warshalls and Johnsons. Execute your programs for the following sets of data: Set_1 Connectivity of each vertex is 10 Set_2 Connectivity is |V|/2 for each vertex Set_3 A Fully connected graph Set_4 A Ring Presentation of Results: Measure CPU time to find the all pairs shortest paths in graphs of sizes 100, 1000, and 10K. Present your results using tables or graphs and write a 1-page report. Summarize the behavior of the two algorithms. OR Write a program to extend the Rabin-Karp method to handle the problem of looking for a given m ( m pattern in an n ( n array of characters. (The pattern may be shifted vertically or horizontally, but it may not be rotated). Use another method, for example, a variation of the KMP algorithm or a method from a research paper, or your own method and compare the results with those of the Rabin-Karp extension. Set_1 n = 64, m= 4, 8 and 16 Set_2 n = 256, m =4, 16, and 64 Set_3 n = 1024, m= 4, 16, and 64 Presentation of Results: Measure CPU time, compare results and present using tables or plots. PLEASE NOTE THAT YOUR PROGRAMS WILL BE TESTED WITH DIFFERENT DATA SETS AS WELL +BCDSghrsxy    - 1 7 A θ|umcXPEhrW>hrW>CJaJhrW>CJaJh*KhrW>CJaJhKzh`56hKzh`6 hKz56 h656h|$h|$56hv=56B*phh|$h`56h`56B*phh 56B*phh656B*phhwF56B*phh`h`56B*phh`hwF56 h`56h`h`56h`h`5+CDy  - 2 ] GH ^`gdgd6$a$gd6 & FgdrW>$a$gdrW>$a$gd6$a$gd`$a$gdwF_A L : Y [ \ ] ! ' / o w x y   ! # Fηβww! jhhAh66OJQJ^JhhAh66OJQJ^JhhAh6OJQJ^Jh6OJQJ^J h`6 h66h*Kh6B*CJaJphh6CJaJh*Kh6CJaJh*Kh65CJaJh1CJaJhrW>CJaJh*KhrW>CJaJ.Hg  ]^_gdv=gd|$gd+I'$a$gd6gd6 & F ^gd6   #AGHIJT\]^_Ƶ}v hN<h2h|$CJaJ hwFhKz6B*CJaJph hwFh*K6B*CJaJphh@D+6B*CJaJph hwFh+I'6B*CJaJphh6B*CJaJphhKzh66hhAh65hhAh6OJQJ^JhhAh65OJQJ^J 1h/ =!"#$%D@D NormalCJ_HaJmH nHsH tHDAD Default Paragraph FontRiR  Table Normal4 l4a (k(No List_+CDy-2]GHg  ]^a0@0@0@0@0@0@0@0@0@0 0 0 0 0@0@00@0@0@0 @0 0 0 0@0@0 @0@0 0 @0@0 A _ H_ _ : haxa8*urn:schemas-microsoft-com:office:smarttagsdate 1112004DayMonthYear7Au#a a33xHa#a Mohan Kumarjoseph3Jx9>^`o(. ^`hH.  L ^ `LhH.   ^ `hH. xx^x`hH. HLH^H`LhH. ^`hH. ^`hH. L^`LhH.3Jhk        v=+I'@D+2N<rW>*KKzwF` 16|$@##4##_`@UnknownGz Times New Roman5Symbol3& z Arial;|i0Batang"qhGfGfF F 24d\\ 3QH)?`*CSE 5311 Design and Analysis of Algorithms Mohan Kumarjoseph Oh+'0 $0 L X dpx+CSE 5311 Design and Analysis of AlgorithmsoSE  Mohan KumarohaohaNormalujosephu2seMicrosoft Word 10.0@@{@{F՜.+,0 hp  CSE@UTA \ +CSE 5311 Design and Analysis of Algorithms Title  !"#$%&()*+,-.1Root Entry F0E3Data 1TableWordDocument"SummaryInformation(DocumentSummaryInformation8'CompObjj  FMicrosoft Word Document MSWordDocWord.Document.89qcse5311-old/LabAssignments.htm010064400020320000200000000045001012437722000165630ustar00cse5311cse531100000400000024 Lab Assignments
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Lab Assignment

 
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/Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H2e ôi m endstream endobj 81 0 obj << /Type /XObject /Subtype /Image /Width 42 /Height 1 /BitsPerComponent 8 /ColorSpace 945 0 R /Length 22 /Filter /FlateDecode >> stream Hb` YXP |9 endstream endobj 82 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H```@` endstream endobj 83 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 2 /BitsPerComponent 8 /ColorSpace 944 0 R /Length 43 /Filter /FlateDecode >> stream Hb`db,l("\였Y ; endstream endobj 84 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H##'@' endstream endobj 85 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 2 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 35 /Filter /FlateDecode >> stream Hzӧ߽{  B endstream endobj 86 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H}6ΡC #g\ endstream endobj 87 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HzUJJ Ν; 9 endstream endobj 88 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H{vv6Cuu5@, endstream endobj 89 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hڲe ;  endstream endobj 90 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HzmYY < endstream endobj 91 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream HGff&üy - endstream endobj 92 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 2 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 37 /Filter /FlateDecode >> stream H4i777??3g~0w\{3 H endstream endobj 93 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H0a ~`' endstream endobj 94 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H```x=@2 endstream endobj 95 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H266f``8q"@7M endstream endobj 96 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H={6Coo/@ endstream endobj 97 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HZt)'''@P/ endstream endobj 98 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 2 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream Hbgg޼y3@&%0 endstream endobj 99 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream H`aa0A{ endstream endobj 100 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H:rCaa!@ endstream endobj 101 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 2 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 36 /Filter /FlateDecode >> stream Hڳg޽\QQ`] < endstream endobj 102 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hڹs'W " endstream endobj 103 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H|CHHȽ{ 2 endstream endobj 104 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hz%Ü9s $ endstream endobj 105 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H2e ӧO V endstream endobj 106 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HpႭ-۷ 3 endstream endobj 107 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H}#Ñ#G & endstream endobj 108 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 2 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 35 /Filter /FlateDecode >> stream Ha``055}Çmmmܳgt^ K endstream endobj 109 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 2 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 32 /Filter /FlateDecode >> stream HJLLd``PTT}Ǐ \l> endstream endobj 110 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hb``8{,@G endstream endobj 111 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hc``?s @K endstream endobj 112 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream HJHH```0  endstream endobj 113 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 2 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 35 /Filter /FlateDecode >> stream H֭[}VZZ  H endstream endobj 114 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hz  $ endstream endobj 115 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hsss3ТE = endstream endobj 116 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream Hjmme``HKK0 endstream endobj 117 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H:z(ЬY T endstream endobj 118 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HԩSyxx֮] `@ endstream endobj 119 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 2 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 36 /Filter /FlateDecode >> stream Hjhh```hiiۊ+XXXv ` f endstream endobj 120 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream H*..f``hjj0  endstream endobj 121 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HZ|(C}}Ǐ +A endstream endobj 122 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream HZz<ŋ  endstream endobj 123 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 2 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 31 /Filter /FlateDecode >> stream Hruue``p͛'M[G endstream endobj 124 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H@| endstream endobj 125 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H:~xWW@ endstream endobj 126 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HZb//ݻw  endstream endobj 127 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream HZp=@  endstream endobj 128 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream Hu@j7 endstream endobj 129 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 49 /Filter /FlateDecode >> stream HZfqKKK}}۷oϞ=cbb{;C endstream endobj 130 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hz}[[ F8 endstream endobj 131 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 26 /Filter /FlateDecode >> stream Hs玒)] ?p endstream endobj 132 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream H:s挙YbbbzzIwwׯ_ j { endstream endobj 133 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H:sLPPIcc7o >> endstream endobj 134 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 51 /Filter /FlateDecode >> stream H$󏏏444nnn$$$PPP;;; ~1 endstream endobj 135 0 obj << /Type /XObject /Subtype /Image /Width 42 /Height 1 /BitsPerComponent 8 /ColorSpace 946 0 R /Length 20 /Filter /FlateDecode >> stream Hb`db&0 z endstream endobj 136 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 51 /Filter /FlateDecode >> stream H$bbb<<> stream Hjll,))9~)S<<<>~NGG`7j endstream endobj 138 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 34 /Filter /FlateDecode >> stream Hscc˞={N:r  endstream endobj 139 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H;qDiiiee%@;F endstream endobj 140 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 51 /Filter /FlateDecode >> stream H$xxxģ"""```zzz : endstream endobj 141 0 obj << /Type /XObject /Subtype /Image /Width 42 /Height 1 /BitsPerComponent 8 /ColorSpace 947 0 R /Length 20 /Filter /FlateDecode >> stream Hb`db&0 z endstream endobj 142 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 22 /Filter /FlateDecode >> stream H˗/_`o endstream endobj 143 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H?@x endstream endobj 144 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HzⅺǏ  endstream endobj 145 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H8qbLL @  endstream endobj 146 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H> stream H჆ K endstream endobj 148 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hڰa0xׯ_ { endstream endobj 149 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H7o^bb"@  endstream endobj 150 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H> stream H|9@1 endstream endobj 152 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H3gNPP@  endstream endobj 153 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream HJOO0 ? endstream endobj 154 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 40 /Filter /FlateDecode >> stream H񣾾ϟ> stream Hb`db&0 z endstream endobj 156 0 obj << /Type /XObject /Subtype /Image /Width 15 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 43 /Filter /FlateDecode >> stream Hzb̙͛3eʔiӦ͜9^# endstream endobj 157 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 31 /Filter /FlateDecode >> stream Hz͡C###;q endstream endobj 158 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream HcHHH[[֭[+** j_ endstream endobj 159 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H쬨غuk[[[ppϟ ] endstream endobj 160 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 31 /Filter /FlateDecode >> stream H:tP``fϞ{nd> f endstream endobj 161 0 obj << /Type /XObject /Subtype /Image /Width 13 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 54 /Filter /FlateDecode >> stream H'ggg???+++VVVxxx  endstream endobj 162 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 51 /Filter /FlateDecode >> stream H$uuuiii zxN endstream endobj 163 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H4i̙3 O endstream endobj 164 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hz%)@[ endstream endobj 165 0 obj << /Type /XObject /Subtype /Image /Width 18 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 39 /Filter /FlateDecode >> stream Hq˗/+WXbܹ1y e& endstream endobj 166 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 26 /Filter /FlateDecode >> stream H:r䈋=؜9s H endstream endobj 167 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 31 /Filter /FlateDecode >> stream H mmm'Mt4 endstream endobj 168 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 31 /Filter /FlateDecode >> stream H lmm#"".^j endstream endobj 169 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 31 /Filter /FlateDecode >> stream Hĉmmmn7h4 8 endstream endobj 170 0 obj << /Type /XObject /Subtype /Image /Width 13 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 51 /Filter /FlateDecode >> stream H:wMf̘>}޽{?~0 endstream endobj 171 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 49 /Filter /FlateDecode >> stream HGmoooxxxDDį_=蘒ritu~ endstream endobj 172 0 obj << /Type /XObject /Subtype /Image /Width 18 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 44 /Filter /FlateDecode >> stream H:vǏ'N=zt…- ,Y G& endstream endobj 173 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hz))@O endstream endobj 174 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H4i  y endstream endobj 175 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hĉ`ΝQ endstream endobj 176 0 obj << /Type /XObject /Subtype /Image /Width 19 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 42 /Filter /FlateDecode >> stream H˾}vc{n߾}ٲe+VX "W0Y(6 endstream endobj 177 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 23 /Filter /FlateDecode >> stream Ht''g] 7 endstream endobj 178 0 obj << /Type /XObject /Subtype /Image /Width 18 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 39 /Filter /FlateDecode >> stream Hsζm6[-[6mڷo@!k& endstream endobj 179 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H:{-@ w endstream endobj 180 0 obj << /Type /XObject /Subtype /Image /Width 18 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 40 /Filter /FlateDecode >> stream H:vիaknܸQ/^`% endstream endobj 181 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H:{@ Y endstream endobj 182 0 obj << /Type /XObject /Subtype /Image /Width 19 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 40 /Filter /FlateDecode >> stream Hڸqŋc%KڵԩS0ӧ_z`Xj( endstream endobj 183 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H~êU~`Mm F endstream endobj 184 0 obj << /Type /XObject /Subtype /Image /Width 20 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 43 /Filter /FlateDecode >> stream HgΜ9'N\rz? ]+ endstream endobj 185 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream HĞppp endstream endobj 186 0 obj << /Type /XObject /Subtype /Image /Width 26 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 44 /Filter /FlateDecode >> stream H;3N: L6m̙ׯ_w]l ?: endstream endobj 187 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream H*(((++0 Z endstream endobj 188 0 obj << /Type /XObject /Subtype /Image /Width 23 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 43 /Filter /FlateDecode >> stream Hӧg̘1L>}/W^J5 endstream endobj 189 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream HҰhhh w ' endstream endobj 190 0 obj << /Type /XObject /Subtype /Image /Width 15 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 39 /Filter /FlateDecode >> stream HZpԩS'3vӧ(7$y endstream endobj 191 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream HzMBBي+~ `Zu  endstream endobj 192 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream HO$`t endstream endobj 193 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H )@'| endstream endobj 194 0 obj << /Type /XObject /Subtype /Image /Width 42 /Height 3 /BitsPerComponent 8 /ColorSpace 949 0 R /Length 45 /Filter /FlateDecode >> stream Hb`dbF,lB DNF.n4ȋ.`dXe endstream endobj 195 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H4iܹs g endstream endobj 196 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hr C||<@# endstream endobj 197 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H)@p endstream endobj 198 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 25 /Filter /FlateDecode >> stream H:w޽{w};  endstream endobj 199 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H:uCtt4@p endstream endobj 200 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H;)@o endstream endobj 201 0 obj << /Type /XObject /Subtype /Image /Width 11 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 38 /Filter /FlateDecode >> stream H:ym۶lٲqCHغu  endstream endobj 202 0 obj << /Type /XObject /Subtype /Image /Width 13 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 43 /Filter /FlateDecode >> stream H:p>ػw׭[z`0gΜr* endstream endobj 203 0 obj << /Type /XObject /Subtype /Image /Width 16 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 45 /Filter /FlateDecode >> stream H+ իW  wɒ%gϞ;p@dr endstream endobj 204 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H:uT``U P endstream endobj 205 0 obj << /Type /XObject /Subtype /Image /Width 14 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 48 /Filter /FlateDecode >> stream Hzљ3gN]<==ϟ?w9`d̛7ݻwYO endstream endobj 206 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream HzⅣCUU@F; endstream endobj 207 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H|M{{;@ endstream endobj 208 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H4iٳ [ endstream endobj 209 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream HǧO\\\` !!˗ endstream endobj 210 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hz@[ endstream endobj 211 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H:riiiYǏF) endstream endobj 212 0 obj << /Type /XObject /Subtype /Image /Width 19 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 45 /Filter /FlateDecode >> stream HzÇ}YfMSΜ9Q@b- endstream endobj 213 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 44 /Filter /FlateDecode >> stream H1cÌ3 EEE\;;G`ϫP endstream endobj 214 0 obj << /Type /XObject /Subtype /Image /Width 20 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 51 /Filter /FlateDecode >> stream H׫W^"'OL>}ʔ)a`̙Ǐ{? `0 endstream endobj 215 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 51 /Filter /FlateDecode >> stream H$555  endstream endobj 216 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 41 /Filter /FlateDecode >> stream H{n]]㓒244|o endstream endobj 217 0 obj << /Type /XObject /Subtype /Image /Width 19 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 45 /Filter /FlateDecode >> stream Hg$˗ݻwϜ9s" L6m…Ϟ={\K- endstream endobj 218 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 36 /Filter /FlateDecode >> stream Heee@FFf˖-̖ endstream endobj 219 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream HpӧORRRN0 endstream endobj 220 0 obj << /Type /XObject /Subtype /Image /Width 19 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 53 /Filter /FlateDecode >> stream H//M6u)`d̜9ĉw޽ @W O- endstream endobj 221 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 29 /Filter /FlateDecode >> stream Hg==ӧO?{ >x endstream endobj 222 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 25 /Filter /FlateDecode >> stream Hzݻ>3A endstream endobj 223 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream Hٿ h endstream endobj 224 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hrǿ~ `%P endstream endobj 225 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream Hz #e endstream endobj 226 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HzѣG>|`'B endstream endobj 227 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 35 /Filter /FlateDecode >> stream H:yիYXX_WWȘ5k@v# endstream endobj 228 0 obj << /Type /XObject /Subtype /Image /Width 13 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 40 /Filter /FlateDecode >> stream Hڼyyf ֬Ysҥ0p…/ / endstream endobj 229 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 32 /Filter /FlateDecode >> stream HJIIaaaa;vz @ endstream endobj 230 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H:lYY@%2 endstream endobj 231 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 34 /Filter /FlateDecode >> stream H:x`qqׯ?}tqT - endstream endobj 232 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hjoo9p@ endstream endobj 233 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H6m@  endstream endobj 234 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hznll,@ endstream endobj 235 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream HDZc 2; endstream endobj 236 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 36 /Filter /FlateDecode >> stream H|0f𨨨e`N> ]G endstream endobj 237 0 obj << /Type /XObject /Subtype /Image /Width 14 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 42 /Filter /FlateDecode >> stream HzeE=zN8 y endstream endobj 238 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 32 /Filter /FlateDecode >> stream H#5,r endstream endobj 239 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream Hjkk0 P endstream endobj 240 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HȘ7o  endstream endobj 241 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream Hs玾>@" endstream endobj 242 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream Hڴi@ endstream endobj 243 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H|9s  N endstream endobj 244 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H;wntt4@ [ endstream endobj 245 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H@]^ endstream endobj 246 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 29 /Filter /FlateDecode >> stream Hwބ ` .. p> endstream endobj 247 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H8q"âE m endstream endobj 248 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H}6Crr2@ endstream endobj 249 0 obj << /Type /XObject /Subtype /Image /Width 11 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 38 /Filter /FlateDecode >> stream H'ӧO `ȝ>}:@ M endstream endobj 250 0 obj << /Type /XObject /Subtype /Image /Width 13 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 40 /Filter /FlateDecode >> stream Hڲe˪UVu6lذo߾0{#GU endstream endobj 251 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream Hjll0  endstream endobj 252 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 28 /Filter /FlateDecode >> stream Hի׬YZ \ endstream endobj 253 0 obj << /Type /XObject /Subtype /Image /Width 11 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 42 /Filter /FlateDecode >> stream HڻwfHHHcccssg %%%`֭  endstream endobj 254 0 obj << /Type /XObject /Subtype /Image /Width 11 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 48 /Filter /FlateDecode >> stream H!㍍  endstream endobj 255 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 31 /Filter /FlateDecode >> stream HK-Zdjjg| endstream endobj 256 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 45 /Filter /FlateDecode >> stream H˦~~~烃 /- endstream endobj 257 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 49 /Filter /FlateDecode >> stream H+ׯ_?qիW^^^/nmm=p  endstream endobj 258 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hjhh/--^|9@. endstream endobj 259 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 32 /Filter /FlateDecode >> stream H{3APPPKKK{{;@ endstream endobj 260 0 obj << /Type /XObject /Subtype /Image /Width 42 /Height 1 /BitsPerComponent 8 /ColorSpace 950 0 R /Length 26 /Filter /FlateDecode >> stream Hb`dbF,Bl  endstream endobj 261 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 29 /Filter /FlateDecode >> stream Hxb 3A^^ޔ)S Y endstream endobj 262 0 obj << /Type /XObject /Subtype /Image /Width 14 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 42 /Filter /FlateDecode >> stream H{;6m6mڴnݺׯ۷o U endstream endobj 263 0 obj << /Type /XObject /Subtype /Image /Width 11 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 42 /Filter /FlateDecode >> stream Hھ}SAAAyyǏ333 ` endstream endobj 264 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H2eJYYY}}}]]ݴi >[ endstream endobj 265 0 obj << /Type /XObject /Subtype /Image /Width 11 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 46 /Filter /FlateDecode >> stream Hzʹiˏ;b ^^ޢLG` endstream endobj 266 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream HzԩSccc+**O< `b endstream endobj 267 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 43 /Filter /FlateDecode >> stream Hڰa\aaE &OJ???"e endstream endobj 268 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 49 /Filter /FlateDecode >> stream H5kVAAANNNqq177ŋwQSS[XXx50V endstream endobj 269 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H*//.((X`@* endstream endobj 270 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 32 /Filter /FlateDecode >> stream HNIIeWWYfy g endstream endobj 271 0 obj << /Type /XObject /Subtype /Image /Width 42 /Height 1 /BitsPerComponent 8 /ColorSpace 951 0 R /Length 16 /Filter /FlateDecode >> stream Hb`$0+ endstream endobj 272 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream HڻwY888 999%%eŊkk endstream endobj 273 0 obj << /Type /XObject /Subtype /Image /Width 13 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 37 /Filter /FlateDecode >> stream Hڷo׬Yjժ%0d 1 p endstream endobj 274 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream Hjjj0 & endstream endobj 275 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H26 333 endstream endobj 276 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream HJHHXd @ ~ endstream endobj 277 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream Hw-@vR endstream endobj 278 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H׶mJJJ455͛`n endstream endobj 279 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H[x1@z endstream endobj 280 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H:y򤩩)@  endstream endobj 281 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 31 /Filter /FlateDecode >> stream H2eJrrgn endstream endobj 282 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H@k endstream endobj 283 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream HZrePP@  endstream endobj 284 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream HrppXz5@ endstream endobj 285 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H:x𠩩)@  endstream endobj 286 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 29 /Filter /FlateDecode >> stream HZdϟ?d_ np endstream endobj 287 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H훹9@ endstream endobj 288 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 29 /Filter /FlateDecode >> stream HjjjaUU`C` endstream endobj 289 0 obj << /Type /XObject /Subtype /Image /Width 42 /Height 1 /BitsPerComponent 8 /ColorSpace 952 0 R /Length 16 /Filter /FlateDecode >> stream Hb`$0+ endstream endobj 290 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream Hڷoߜ9s`@PPoΝb^W endstream endobj 291 0 obj << /Type /XObject /Subtype /Image /Width 14 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 40 /Filter /FlateDecode >> stream HwٳgOwޅ ΁sϟ? endstream endobj 292 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream Hjii0 J endstream endobj 293 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HIOO_dO:`6Y endstream endobj 294 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H={6@ endstream endobj 295 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 31 /Filter /FlateDecode >> stream HqㆆƧO> stream HrJBBŒ3&Oliih"b endstream endobj 297 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream Hu@@ endstream endobj 298 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream Hz𡅅@L endstream endobj 299 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 31 /Filter /FlateDecode >> stream H1cFbb/_ܹcffvQj M endstream endobj 300 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream Hߴi@ endstream endobj 301 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H?~HH@ O endstream endobj 302 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H:uWXahh`B$# endstream endobj 303 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H7o^\\@ | endstream endobj 304 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H Zj@ endstream endobj 305 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 31 /Filter /FlateDecode >> stream Hz捚Ǐ߼y:m4t endstream endobj 306 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 36 /Filter /FlateDecode >> stream HZp%o޼׮]0pn 3 endstream endobj 307 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream H:r[ZZ@LL s endstream endobj 308 0 obj << /Type /XObject /Subtype /Image /Width 14 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 42 /Filter /FlateDecode >> stream Hׯk׀ f͚5 ͛w  endstream endobj 309 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 29 /Filter /FlateDecode >> stream H:w8o>@ endstream endobj 310 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HڲeKRR_LLڵk :H endstream endobj 311 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 49 /Filter /FlateDecode >> stream H;wׇxQ=~ endstream endobj 312 0 obj << /Type /XObject /Subtype /Image /Width 20 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 73 /Filter /FlateDecode >> stream HzyZZGiiiDD͛7[ZZ^~allonnnddc۬^ endstream endobj 313 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream HZzǏ ;v endstream endobj 314 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 29 /Filter /FlateDecode >> stream HrJC endstream endobj 315 0 obj << /Type /XObject /Subtype /Image /Width 13 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 54 /Filter /FlateDecode >> stream H'ZZZ@@@ddd:::,,,000 Hf endstream endobj 316 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H*--uuuܱc@.e endstream endobj 317 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 37 /Filter /FlateDecode >> stream Hnݪ,,,k֬~:@] endstream endobj 318 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 36 /Filter /FlateDecode >> stream HrJQQ=ϟ endstream endobj 319 0 obj << /Type /XObject /Subtype /Image /Width 19 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 51 /Filter /FlateDecode >> stream HzÇǏw={00mڴs~`0?- endstream endobj 320 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream HѣGϜ9sY ͛7 endstream endobj 321 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H iu endstream endobj 322 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream H * endstream endobj 323 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hz0 j/ endstream endobj 324 0 obj << /Type /XObject /Subtype /Image /Width 20 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 63 /Filter /FlateDecode >> stream H QT4M80 6[nfE2sw}w3#3 endstream endobj 325 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream H ~ endstream endobj 326 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream HѣGO:uy ӧ endstream endobj 327 0 obj << /Type /XObject /Subtype /Image /Width 13 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 54 /Filter /FlateDecode >> stream H'е777 O endstream endobj 328 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H0x j endstream endobj 329 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 39 /Filter /FlateDecode >> stream HzݥKYXXzzz@ endstream endobj 330 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 43 /Filter /FlateDecode >> stream HǓ'Oyyy_|ӹ x endstream endobj 331 0 obj << /Type /XObject /Subtype /Image /Width 17 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 52 /Filter /FlateDecode >> stream H7o^ׯ>|8eʔɓ'O3f;w޽<(r endstream endobj 332 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H˳Zh@ endstream endobj 333 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 39 /Filter /FlateDecode >> stream Hz۷oݺrrr>}0 H endstream endobj 334 0 obj << /Type /XObject /Subtype /Image /Width 11 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 42 /Filter /FlateDecode >> stream HקO>|`oo/&&x `A endstream endobj 335 0 obj << /Type /XObject /Subtype /Image /Width 11 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 36 /Filter /FlateDecode >> stream Hg$˗͛7Ϙ1cʔ)@g endstream endobj 336 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 25 /Filter /FlateDecode >> stream HzW>}H _ endstream endobj 337 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 29 /Filter /FlateDecode >> stream HzS<==`@[[;""  endstream endobj 338 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 29 /Filter /FlateDecode >> stream Hz왕 :u ]?x endstream endobj 339 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 15 /Filter /FlateDecode >> stream H  ,( endstream endobj 340 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HzՑ#G={`& endstream endobj 341 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream Hzͽ{ _ endstream endobj 342 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream Hھbcc@[[5(( endstream endobj 343 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 32 /Filter /FlateDecode >> stream H~ر hy) endstream endobj 344 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H:zjyy9@v$ endstream endobj 345 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream Hӱc ) endstream endobj 346 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H6m'@  endstream endobj 347 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream HsΕ+W  endstream endobj 348 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream HZf͢E*++`@SS3&&Ν; y endstream endobj 349 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream Hھ}{LLtww/Zhd  endstream endobj 350 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream Hjjj*..0 V endstream endobj 351 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H{@ " endstream endobj 352 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream Hrܹs  T endstream endobj 353 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream Hjoo0 e endstream endobj 354 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 29 /Filter /FlateDecode >> stream HZzu[[ S p endstream endobj 355 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 32 /Filter /FlateDecode >> stream HZbEAA pss޽ V(Q endstream endobj 356 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 48 /Filter /FlateDecode >> stream H׮]ښ:::nݺjժ߿0E6 endstream endobj 357 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H :uj[[۝;w`[ z endstream endobj 358 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 23 /Filter /FlateDecode >> stream HΖ͛7+w| endstream endobj 359 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 49 /Filter /FlateDecode >> stream HZ~}GGǙ3g,XPSS3{?x𠫫@.@XW endstream endobj 360 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 28 /Filter /FlateDecode >> stream Hdž jkk-[Z J endstream endobj 361 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 28 /Filter /FlateDecode >> stream H2VZUWWcǎhR  endstream endobj 362 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 28 /Filter /FlateDecode >> stream H'L+\ , endstream endobj 363 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream HzmCCCgg' 7P; endstream endobj 364 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream HZ~}wwmۚ{{{=z`k endstream endobj 365 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 26 /Filter /FlateDecode >> stream Hv횝a%ܽ{ R = endstream endobj 366 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 28 /Filter /FlateDecode >> stream Hŋ6oY7 S endstream endobj 367 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 47 /Filter /FlateDecode >> stream Hڻwo[[ۉ'VXQ[[;y߿@bz endstream endobj 368 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 32 /Filter /FlateDecode >> stream Hڷo_prr2ACCC[[͛ N Q endstream endobj 369 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 32 /Filter /FlateDecode >> stream HZhQ[[[II HIIUTT;v pb endstream endobj 370 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 44 /Filter /FlateDecode >> stream H:|\\\yyyܹsgٲe\\\@ϟ c endstream endobj 371 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H1cFXXX}}{{ Q  endstream endobj 372 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 23 /Filter /FlateDecode >> stream HJMM( 6)74 endstream endobj 373 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 51 /Filter /FlateDecode >> stream H$yyysss䖖pppzzz W& endstream endobj 374 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H9sfMMMCCCeee__@? endstream endobj 375 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H+((jkk۰a@CS endstream endobj 376 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 42 /Filter /FlateDecode >> stream HڼysGGGfffHHHSSׯ_߼ySYYikk` endstream endobj 377 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream HZj`aaaEEEvvvRR@GT endstream endobj 378 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 26 /Filter /FlateDecode >> stream H|~ܹs P4 endstream endobj 379 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hilllkk?>@> stream Hڹsjffˋz{{߿!Q endstream endobj 381 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 29 /Filter /FlateDecode >> stream H:{lRR 477X `No endstream endobj 382 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 32 /Filter /FlateDecode >> stream H?~AA%&&&%%=~ ^A endstream endobj 383 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HuVMMM=ڵׯ_l  endstream endobj 384 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H{.@I endstream endobj 385 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H}@OL endstream endobj 386 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H= `m endstream endobj 387 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H-@U endstream endobj 388 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H joo@~z endstream endobj 389 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H:uꔸx@@@RR̙3 E endstream endobj 390 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 26 /Filter /FlateDecode >> stream HMKK0 endstream endobj 391 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream Hսu@ endstream endobj 392 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H9sfXX@  endstream endobj 393 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream HJKKkmm۷o>|`i endstream endobj 394 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream Hv횞@  endstream endobj 395 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream Hrss5k@ endstream endobj 396 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream H0  endstream endobj 397 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 25 /Filter /FlateDecode >> stream HRUULKK endstream endobj 398 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H,((@5n endstream endobj 399 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream Hz]aaauuuCCúu|`m endstream endobj 400 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 32 /Filter /FlateDecode >> stream Hz捯n l endstream endobj 401 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream Hƍ`(((` endstream endobj 402 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream H涶{{{ . ~ endstream endobj 403 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H۳g;;; A: endstream endobj 404 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 31 /Filter /FlateDecode >> stream H0aB||o> stream Hy󦆆ƧO?~\YYzjqp endstream endobj 406 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream Hrss;x @ endstream endobj 407 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 31 /Filter /FlateDecode >> stream H:y򤹹yoo5kvc endstream endobj 408 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HrqqIOO_t;N8`7nq endstream endobj 409 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H񣺺`p  endstream endobj 410 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream HɩϹs >} `f! \ endstream endobj 411 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H7o^pp0@ : endstream endobj 412 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H;}4@j endstream endobj 413 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream Hzꕉ @[ endstream endobj 414 0 obj << /Type /XObject /Subtype /Image /Width 11 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 48 /Filter /FlateDecode >> stream H!bbbaaa:::ppp S`S endstream endobj 415 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream HXd @  endstream endobj 416 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream H ` endstream endobj 417 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream H㇟6 DEE>|˗/Z endstream endobj 418 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 29 /Filter /FlateDecode >> stream Ha0ppp`==P endstream endobj 419 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream HtRLLLRRg\\ŋ V  endstream endobj 420 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 48 /Filter /FlateDecode >> stream HurrrssTQQ000={o 6  endstream endobj 421 0 obj << /Type /XObject /Subtype /Image /Width 11 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 45 /Filter /FlateDecode >> stream HKffgCCCvv5??˗/ T endstream endobj 422 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hٳ}͛f W endstream endobj 423 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HڳgOzzODDIJe > stream H󧪪jrruzz3g T endstream endobj 425 0 obj << /Type /XObject /Subtype /Image /Width 19 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 66 /Filter /FlateDecode >> stream HvZ]]Y~S [YYYHHؾ};P600qㆧ'_II @ a endstream endobj 426 0 obj << /Type /XObject /Subtype /Image /Width 17 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 63 /Filter /FlateDecode >> stream H|rrrOPPP{{۷o7o ILL> stream HzEdddZZWTT֭[ Z~  endstream endobj 428 0 obj << /Type /XObject /Subtype /Image /Width 11 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 35 /Filter /FlateDecode >> stream HUTT*| 46 endstream endobj 429 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 36 /Filter /FlateDecode >> stream HzիW@QQQEE%??ׯ endstream endobj 430 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HY08wܣG ` endstream endobj 431 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 47 /Filter /FlateDecode >> stream H Ʊ _J:ytT3vfFDwK?ə`6 endstream endobj 432 0 obj << /Type /XObject /Subtype /Image /Width 11 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 44 /Filter /FlateDecode >> stream H﯂?~ر/>}ĉ?0 endstream endobj 433 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream H 2 endstream endobj 434 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H j# endstream endobj 435 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hz𡚚^0x pn ' endstream endobj 436 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H l x endstream endobj 437 0 obj << /Type /XObject /Subtype /Image /Width 13 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 49 /Filter /FlateDecode >> stream H ȡ CM!%XpMw7"dcfϪt endstream endobj 438 0 obj << /Type /XObject /Subtype /Image /Width 17 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 58 /Filter /FlateDecode >> stream H1Q_ f]@'Jﷻ."YUE͌f 0>, endstream endobj 439 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream H  endstream endobj 440 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 42 /Filter /FlateDecode >> stream HzeWW???33377Ɲ;w޽{`b endstream endobj 441 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 39 /Filter /FlateDecode >> stream HzוDDDZZZ^| `R W endstream endobj 442 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 23 /Filter /FlateDecode >> stream H0I endstream endobj 443 0 obj << /Type /XObject /Subtype /Image /Width 11 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 48 /Filter /FlateDecode >> stream H!ڞ###  endstream endobj 444 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 42 /Filter /FlateDecode >> stream HOvޭ$g̘q֭۷oBX endstream endobj 445 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 25 /Filter /FlateDecode >> stream Hz^z{H ) endstream endobj 446 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream Hxm888d…k9 endstream endobj 447 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream HZr lܸ… J9 endstream endobj 448 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 26 /Filter /FlateDecode >> stream Hf777'  endstream endobj 449 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 26 /Filter /FlateDecode >> stream H6m;39 endstream endobj 450 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream H.((0 endstream endobj 451 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H8qbAAA{{;@[ endstream endobj 452 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 953 0 R /Length 25 /Filter /FlateDecode >> stream Hb`d,̬hl e endstream endobj 453 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 954 0 R /Length 26 /Filter /FlateDecode >> stream Hb`DLLL,̌X+@> endstream endobj 454 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 955 0 R /Length 16 /Filter /FlateDecode >> stream Hb`$0, endstream endobj 455 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 956 0 R /Length 17 /Filter /FlateDecode >> stream Hb`$01. endstream endobj 456 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 957 0 R /Length 20 /Filter /FlateDecode >> stream Hb`db& M endstream endobj 457 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 958 0 R /Length 20 /Filter /FlateDecode >> stream Hb`db& M endstream endobj 458 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hz ##>@RU endstream endobj 459 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HJJJb``8q@  endstream endobj 460 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HڷoCQQ@ endstream endobj 461 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HRVV @ j endstream endobj 462 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hz @OO endstream endobj 463 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hf``8q@ Aj endstream endobj 464 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HڷoCVV@Uy endstream endobj 465 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HRPP @ d endstream endobj 466 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream H b endstream endobj 467 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream H \ endstream endobj 468 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream HpBKK~rr US endstream endobj 469 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream HzqVVބ }`WB endstream endobj 470 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H>}ӧO^QQ`qy D endstream endobj 471 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H }{zz de endstream endobj 472 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H˗~ `=; endstream endobj 473 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H2008{,@ endstream endobj 474 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HY""" # endstream endobj 475 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H:pԩSSSS / endstream endobj 476 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream Hjmm^`E||| [ endstream endobj 477 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H7336m@  endstream endobj 478 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 26 /Filter /FlateDecode >> stream H󃂂 ** hj % endstream endobj 479 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream Hڰa@ j endstream endobj 480 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H_yyy{{;@! endstream endobj 481 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HZrc`!uK endstream endobj 482 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H:q℅@  endstream endobj 483 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hz]UUӓ'O 3` endstream endobj 484 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H5kV}}@@@[[֭[ Q] endstream endobj 485 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hjll HMM=r@5c endstream endobj 486 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream Hʤ 1z endstream endobj 487 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 25 /Filter /FlateDecode >> stream Hڵk… m7  endstream endobj 488 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HJHH```8q@ y endstream endobj 489 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hz ##@LR endstream endobj 490 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HڷoCAA@y endstream endobj 491 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HRRR @ g endstream endobj 492 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 959 0 R /Length 20 /Filter /FlateDecode >> stream Hb`db& M endstream endobj 493 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 960 0 R /Length 20 /Filter /FlateDecode >> stream Hb`db& M endstream endobj 494 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 961 0 R /Length 17 /Filter /FlateDecode >> stream Hb`d"0U endstream endobj 495 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 962 0 R /Length 17 /Filter /FlateDecode >> stream Hb`$01. endstream endobj 496 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 963 0 R /Length 25 /Filter /FlateDecode >> stream Hb`dbF,lb endstream endobj 497 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 964 0 R /Length 28 /Filter /FlateDecode >> stream Hb`dbF,h"l쌌 endstream endobj 498 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream Hbdd4440; endstream endobj 499 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hf``@ U endstream endobj 500 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 26 /Filter /FlateDecode >> stream Hx񢁁HJJ~ 8+ endstream endobj 501 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hz5##@4O endstream endobj 502 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H277g``x)@ U endstream endobj 503 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream HڻwCFFЗ/_ I- endstream endobj 504 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H @ la endstream endobj 505 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 36 /Filter /FlateDecode >> stream HڿCnn_455yxx|`ws N endstream endobj 506 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H:t萜Cnn / endstream endobj 507 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H:~(Cffׯ_ /O endstream endobj 508 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H:v옴4Cvvׯ_ 0  endstream endobj 509 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H:y$Cjj . endstream endobj 510 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H:u???Crr2@ endstream endobj 511 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H:uꔐCZZ@* endstream endobj 512 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H:q℈CJJ @  endstream endobj 513 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H5kVWW׏? _U endstream endobj 514 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H3g޽{  endstream endobj 515 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H:t萤$Cjj*@ endstream endobj 516 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream Hb``0 [ endstream endobj 517 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream HZre[[@  endstream endobj 518 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream H1c > endstream endobj 519 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hڷo,CZZ@ endstream endobj 520 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H[VV֭[ ; endstream endobj 521 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream Hc``HNN0 endstream endobj 522 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H6m+++ôi~`' endstream endobj 523 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hڷo<CFF@ endstream endobj 524 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H; %t endstream endobj 525 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hڷo$CRRү_ . endstream endobj 526 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H:u*̙3  endstream endobj 527 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hڿ"Cvvϟ? / endstream endobj 528 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream Hd``0 u endstream endobj 529 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H:}##_ ! endstream endobj 530 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream Hd`` 0r endstream endobj 531 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H:r䈜Cvvo 0 endstream endobj 532 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hŋ &| endstream endobj 533 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 36 /Filter /FlateDecode >> stream Hz5##ӧ|`{ ? endstream endobj 534 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H1c;;E  endstream endobj 535 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hz .@UR endstream endobj 536 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hd``x-@ s endstream endobj 537 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hz𡚚Cpp? 4 endstream endobj 538 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hc``8q@ s endstream endobj 539 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HڷoC^^@m endstream endobj 540 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H' @(/i endstream endobj 541 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H3///8Caa!@8 endstream endobj 542 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HRTT @ g endstream endobj 543 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 965 0 R /Length 38 /Filter /FlateDecode >> stream Hb`dbfaecg`N6.8H f endstream endobj 544 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 966 0 R /Length 36 /Filter /FlateDecode >> stream Hb`dbfffadecvFFN.f` r endstream endobj 545 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 967 0 R /Length 23 /Filter /FlateDecode >> stream Hb`dbf!Y@ P endstream endobj 546 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 968 0 R /Length 22 /Filter /FlateDecode >> stream Hb`dddbf$ XX /5 endstream endobj 547 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 969 0 R /Length 35 /Filter /FlateDecode >> stream Hb`dbbfaaeC,<&&N6.n endstream endobj 548 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 970 0 R /Length 31 /Filter /FlateDecode >> stream Hb`dbfaeecgFFF `0 | endstream endobj 549 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream HZl;;ĉ A endstream endobj 550 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream HCkk+@%) endstream endobj 551 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H*//g``~9@B endstream endobj 552 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H5k33ɓ  endstream endobj 553 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hڱc !6 endstream endobj 554 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H:w#ѣG %P} endstream endobj 555 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H>}:@1 endstream endobj 556 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hqㆋ åK 4 endstream endobj 557 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hg`` w@m endstream endobj 558 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream Hacc0  endstream endobj 559 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hz𡏏òe (z endstream endobj 560 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H:w9æM #& endstream endobj 561 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream H`e˖x endstream endobj 562 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream He``y&@ endstream endobj 563 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H:uꔎCddÇ 1\ endstream endobj 564 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hjmmeddd``Xr%@? endstream endobj 565 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hw^@@[ 6 endstream endobj 566 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream HZbCtt7o  endstream endobj 567 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hjkkcbbb``Xb@r endstream endobj 568 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hzꕫ+Ç )@ endstream endobj 569 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H:y򤂂Cgg'@ +W endstream endobj 570 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream Hb+V endstream endobj 571 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hb``r @C endstream endobj 572 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hw8CJJ۷o 2 endstream endobj 573 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Ha``X|9@z endstream endobj 574 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HSDD+W 9 endstream endobj 575 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hڱc ! endstream endobj 576 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream He``Yd @ endstream endobj 577 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream Hd``0 f endstream endobj 578 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hڻw/Cii)@ endstream endobj 579 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hc``]`@ endstream endobj 580 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hdcc;~8@ endstream endobj 581 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream Haee-((0 endstream endobj 582 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hkoo/// > endstream endobj 583 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream HJKKc``9x @ endstream endobj 584 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HZv- @\ endstream endobj 585 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hz=Ă  $ endstream endobj 586 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream Hf``0220 = endstream endobj 587 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream Hhll07 endstream endobj 588 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream He``0770  endstream endobj 589 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H3  %; endstream endobj 590 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H;w.'''777@ endstream endobj 591 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hb``]n@ endstream endobj 592 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HKuu5Ȋ+ = endstream endobj 593 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H:u*+++ɓ + endstream endobj 594 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hd``ؾ};@g endstream endobj 595 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream Hc``0 endstream endobj 596 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H211a``>}:@sb endstream endobj 597 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H*//g```eeݾ};@ endstream endobj 598 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream HpĂ   endstream endobj 599 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hjkkc``x @* endstream endobj 600 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H*//g```bbڹs'@ endstream endobj 601 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hs__??+ ?  endstream endobj 602 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Haccc``X|ϟ? 'l endstream endobj 603 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H````cc۾};@( endstream endobj 604 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream Hjjjb``hhh09 endstream endobj 605 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H{\\  ,s endstream endobj 606 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HWuu5϶m = endstream endobj 607 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream HZf ++3  endstream endobj 608 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H4iӧ l endstream endobj 609 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HWmm-Ж-[ >v endstream endobj 610 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H9s&Ì3~ `( endstream endobj 611 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hkzz: - endstream endobj 612 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hg``ٸq#@ endstream endobj 613 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H6m0͛7 @ endstream endobj 614 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HzUaa! ;= endstream endobj 615 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HC]] = endstream endobj 616 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 20 /Filter /FlateDecode >> stream H d``0  endstream endobj 617 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream HZ|9={ D endstream endobj 618 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H:{,ºu m endstream endobj 619 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H?>C[[ ) endstream endobj 620 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HZd Cee%@?X endstream endobj 621 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H:s '@@ endstream endobj 622 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream HZb˗/ : endstream endobj 623 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hzj}}޽{ 4 endstream endobj 624 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream HzIn `" endstream endobj 625 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 2 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 31 /Filter /FlateDecode >> stream HRVVf``xծ]VZ/Fw B endstream endobj 626 0 obj << /Type /XObject /Subtype /Image /Width 41 /Height 1 /BitsPerComponent 8 /ColorSpace 971 0 R /Length 19 /Filter /FlateDecode >> stream Hb`d"0sV endstream endobj 627 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 972 0 R /Length 20 /Filter /FlateDecode >> stream Hb`db& M endstream endobj 628 0 obj << /Type /XObject /Subtype /Image /Width 42 /Height 1 /BitsPerComponent 8 /ColorSpace 973 0 R /Length 20 /Filter /FlateDecode >> stream Hb`db&  endstream endobj 629 0 obj << /Type /XObject /Subtype /Image /Width 41 /Height 1 /BitsPerComponent 8 /ColorSpace 974 0 R /Length 19 /Filter /FlateDecode >> stream Hb``$0108* endstream endobj 630 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 975 0 R /Length 20 /Filter /FlateDecode >> stream Hb`db&01 ?{ endstream endobj 631 0 obj << /Type /XObject /Subtype /Image /Width 42 /Height 1 /BitsPerComponent 8 /ColorSpace 976 0 R /Length 20 /Filter /FlateDecode >> stream Hb`dd"03S endstream endobj 632 0 obj << /Type /XObject /Subtype /Image /Width 41 /Height 1 /BitsPerComponent 8 /ColorSpace 977 0 R /Length 19 /Filter /FlateDecode >> stream Hb`d"0sV endstream endobj 633 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 978 0 R /Length 20 /Filter /FlateDecode >> stream Hb`db& M endstream endobj 634 0 obj << /Type /XObject /Subtype /Image /Width 42 /Height 1 /BitsPerComponent 8 /ColorSpace 979 0 R /Length 20 /Filter /FlateDecode >> stream Hb`db&  endstream endobj 635 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream Hb``0u endstream endobj 636 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream H0 endstream endobj 637 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H //"@rg endstream endobj 638 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream He``طo@ endstream endobj 639 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hڸq#Cee%@w endstream endobj 640 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream H0 endstream endobj 641 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream Hb``0l endstream endobj 642 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H󧈈$@#s endstream endobj 643 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H //@ld endstream endobj 644 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hb``طo@my endstream endobj 645 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hڸq#CYY@ev endstream endobj 646 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H󧈈@p endstream endobj 647 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H};~`%) endstream endobj 648 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 25 /Filter /FlateDecode >> stream Hϟ?;~HW x endstream endobj 649 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 25 /Filter /FlateDecode >> stream H:{%Kf͚uI7} endstream endobj 650 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream HGUU1@18 endstream endobj 651 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream HIIثW 2Z endstream endobj 652 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HZtiAAAtttbbU ;r endstream endobj 653 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H{GGGEE-@5 endstream endobj 654 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hڰa ޼y`- endstream endobj 655 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H:|U`, endstream endobj 656 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream Hx񢵵5@ % endstream endobj 657 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hziii'lmm>} `Pg endstream endobj 658 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream H0@ endstream endobj 659 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H:x𠵵5@  endstream endobj 660 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream HDDD \  endstream endobj 661 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H ޹s =V endstream endobj 662 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 29 /Filter /FlateDecode >> stream HRVVسg… sss$@I endstream endobj 663 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream Hå555   endstream endobj 664 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 18 /Filter /FlateDecode >> stream H:v옭-@  endstream endobj 665 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream H033@A6 endstream endobj 666 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H˗AKKKTT@^ endstream endobj 667 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 28 /Filter /FlateDecode >> stream HzUtuu> stream Hַ d endstream endobj 669 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H*((3g·.\PXX`l > endstream endobj 670 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream Hy?@J endstream endobj 671 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream HJIIЈ7@ endstream endobj 672 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hzivv!̙3ǏY endstream endobj 673 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 26 /Filter /FlateDecode >> stream Ho~ `sqt endstream endobj 674 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hϟ?`+ endstream endobj 675 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hח/_}`+ endstream endobj 676 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream H0 endstream endobj 677 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream Hb``hnn0 endstream endobj 678 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H 2@oj endstream endobj 679 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H***b``طo@- endstream endobj 680 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hڸq#C}}=@ endstream endobj 681 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream H0 endstream endobj 682 0 obj << /Type /XObject /Subtype /Image /Width 41 /Height 1 /BitsPerComponent 8 /ColorSpace 980 0 R /Length 24 /Filter /FlateDecode >> stream Hb`dBL \ endstream endobj 683 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 981 0 R /Length 24 /Filter /FlateDecode >> stream Hb`dbF,,Bl endstream endobj 684 0 obj << /Type /XObject /Subtype /Image /Width 42 /Height 1 /BitsPerComponent 8 /ColorSpace 982 0 R /Length 25 /Filter /FlateDecode >> stream Hb`dbF,,,  endstream endobj 685 0 obj << /Type /XObject /Subtype /Image /Width 41 /Height 1 /BitsPerComponent 8 /ColorSpace 983 0 R /Length 16 /Filter /FlateDecode >> stream Hb` 0* endstream endobj 686 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 984 0 R /Length 17 /Filter /FlateDecode >> stream Hb`d"0U endstream endobj 687 0 obj << /Type /XObject /Subtype /Image /Width 42 /Height 1 /BitsPerComponent 8 /ColorSpace 985 0 R /Length 16 /Filter /FlateDecode >> stream Hb`$0+ endstream endobj 688 0 obj << /Type /XObject /Subtype /Image /Width 41 /Height 1 /BitsPerComponent 8 /ColorSpace 986 0 R /Length 31 /Filter /FlateDecode >> stream Hb`dB, l(BL \Ml endstream endobj 689 0 obj << /Type /XObject /Subtype /Image /Width 43 /Height 1 /BitsPerComponent 8 /ColorSpace 987 0 R /Length 32 /Filter /FlateDecode >> stream Hb`dbF,l(B\< endstream endobj 690 0 obj << /Type /XObject /Subtype /Image /Width 42 /Height 1 /BitsPerComponent 8 /ColorSpace 988 0 R /Length 30 /Filter /FlateDecode >> stream Hb`dbF,l̨ endstream endobj 691 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream Hb``hhh0 endstream endobj 692 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 26 /Filter /FlateDecode >> stream HJMMefffaaa``pqq0 endstream endobj 693 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 26 /Filter /FlateDecode >> stream HjhhbaaaddTRR0 endstream endobj 694 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H @ig endstream endobj 695 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H>} Ӌ/  endstream endobj 696 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream H:z### þ} 1 endstream endobj 697 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hڸq#Cmm-@ endstream endobj 698 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 26 /Filter /FlateDecode >> stream HJKKcfffaaa``pqq0& endstream endobj 699 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 23 /Filter /FlateDecode >> stream H*,,dccc``PTT0 endstream endobj 700 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream HWqq1''ۥ f' endstream endobj 701 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 32 /Filter /FlateDecode >> stream H[XX; b * endstream endobj 702 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 32 /Filter /FlateDecode >> stream Hھ};///CWWW~~>o>N6r endstream endobj 703 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream Hbdddcccgggbb}6@"h` endstream endobj 704 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 30 /Filter /FlateDecode >> stream HGrr277733@C endstream endobj 705 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 32 /Filter /FlateDecode >> stream H[xx8Ν; b4- endstream endobj 706 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HgYY?? = endstream endobj 707 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 35 /Filter /FlateDecode >> stream HSDD{޿d[[[t B endstream endobj 708 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 17 /Filter /FlateDecode >> stream H0 endstream endobj 709 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hڹs'CYY@9 endstream endobj 710 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 32 /Filter /FlateDecode >> stream Ha``ܰa}mEK endstream endobj 711 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H*((```طo@ endstream endobj 712 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 33 /Filter /FlateDecode >> stream HkZZѣGeee%%% e(? endstream endobj 713 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 35 /Filter /FlateDecode >> stream HSdd$Ľ{>|d[[[ E endstream endobj 714 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hz5@X endstream endobj 715 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hz}LL ˗ 9 endstream endobj 716 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hgee%Ν; > endstream endobj 717 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H뗈$@;v endstream endobj 718 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hruue``}6@ a endstream endobj 719 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hc``Yx1@ endstream endobj 720 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hڵk Cii)@$ endstream endobj 721 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hz}LL ܥK 9 endstream endobj 722 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hz15@[ endstream endobj 723 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HsFFӧ ; endstream endobj 724 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hz]\\ŋ :J endstream endobj 725 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hz!5@U endstream endobj 726 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hd``߶m@B endstream endobj 727 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream He``]`@ endstream endobj 728 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream Hrvvf``}6@ s[ endstream endobj 729 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hڵk;;;CYY@r endstream endobj 730 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hz}\\ :Y endstream endobj 731 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 21 /Filter /FlateDecode >> stream H뗈@5s endstream endobj 732 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HkVVѣG > stream He``Y`@ endstream endobj 734 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HC\\ƹs : endstream endobj 735 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H*//g``ؼy3@ endstream endobj 736 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hc||<ƙ3g : endstream endobj 737 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hڹs'CEE@ endstream endobj 738 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hkll,ɓ' ;[ endstream endobj 739 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HGVV < endstream endobj 740 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream He``\`@ endstream endobj 741 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H*++c``۴i@ endstream endobj 742 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream H[||<ɓ' ; endstream endobj 743 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream HڶmCYY@ endstream endobj 744 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream Hgdd$' ;y endstream endobj 745 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H```PTTܿ?@ endstream endobj 746 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream Hf```ee]t)@  endstream endobj 747 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 24 /Filter /FlateDecode >> stream H*--e``޲e @v endstream endobj 748 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 1170 0 R /Length 27 /Filter /FlateDecode >> stream HgTT' ;| 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No work is needed to combine them b^!9 t 3333#3333.3333:33 3333333 N33Example     Running Time of Quicksort&   Mergesort    Example      Runtime Complexity of Mergesort   Runtime Complexity of Mergesort     Questions   uWhat is a pivot element? Do you understand divide-and-conquer? What is running time if pivot element is at the center of the array? How is Mergesort different from Quicksort? Trace the algorithm with the help of an example? What is the use of TEMP array? Do you know why we execute steps 13 and 14 in the MERGE algorithm? What happens if i >q after the while loop (4-11)? HuB 2  Find applications for selection sort, heapsort, quicksort, and mergesort sorting algorithms? Which of these problems are suitable for different types of sorting operations? Identify applications for which each sorting algorithm works bestb&        ?Insertion Sort Counting Sort Find the Maximum Find the Minimum8The Odd-Even Mergesort problem, B 9 18 27 36 45 54 63 72 64 56 48 40 32 24 16 8 Bitonic Mergesort>C 0 1 Sorting Networks  !Shuffle-Exchange Network Week 6"!   Parallel Bitonic Sort Algorithm !!  Given Unsorted list :&  ` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> $(    6PӖ P  T Click to edit Master title style! !  0Ֆ   RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0ܖ ``  X*  0 `   Z*  0 `   Z*H  0޽h ? ̙33 Default Design 0 0*(    0H P    X*   0b     Z* d  c $ ?    0   @  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  6 `P   X*   6 `   Z* H  0޽h ? ̙3380___PPT10.bq5 @(  l  C   l  C l      0 K  3Further Reading Chapters 1 and 8 from Textbook H3 233H  0޽h ? ̙33  P ( v   l  C dJ    l  C 8Kp    H  0޽h ? ̙3354  `O(  l  C @e      0 hp` : 2   0,$  0 m13 02 18 26 76 87 98 11 93 77 65 43 38 09 65 06 13 02 06 26 76 87 98 11 93 77 65 43 38 09 65 18 13 02 06 09 76 87 98 11 93 77 65 43 38 26 65 18 13 02 06 09 11 87 98 76 93 77 65 43 38 26 65 18 11 02 06 09 13 87 98 76 93 77 65 43 38 26 65 18 11 02 06 09 13 87 98 76 93 77 65 43 38 26 65 18 09 02 06 11 13 87 98 76 93 77 65 43 38 26 65 18 09 02 06 11 13 87 98 76 93 77 65 43 38 26 65 18 06 02 09 11 13 87 98 76 93 77 65 43 38 26 65 18 06 02 09 11 13 87 98 76 93 77 65 43 38 26 65 18 02 06 09 11 13 87 98 76 93 77 65 43 38 26 65 18 02 06 09 11 13 87 18 76 93 77 65 43 38 26 65 98 02 06 09 11 13 87 18 76 65 77 65 43 38 26 93 98 02 06 09 11 13 26 18 76 65 77 65 43 38 87 93 98 02 06 09 11 13 18 26 76 65 77 65 43 38 87 93 98 02 06 09 11 13 18 26 76 65 38 65 43 77 87 93 98 02 06 09 11 13 18 26 43 65 38 65 76 77 87 93 98 02 06 09 11 13 18 26 43 38 65 65 76 77 87 93 98 02 06 09 11 13 18 26 38 43 65 65 76 77 87 93 98 02 06 09 11 13 18 26 38 43 65 65 76 77 87 93 98  P 33< 3333!3333! 33f3331 3333f3331 f3f3331  f3f3331  f3f3f3331 f3f3f3331 f3f3f3f3f333( f3f3f3f3f333 f3f3f3f3f333 f3f3f3f3f333 f3f3f3f3f3f333f3f3  f3f3f3f3f3f333f3f3f3f3f3f3f3f333f3f3f3f3f3f3f3f3f333f3f3f3f3f3f3f3f3f333f3f3f3f3f3f3f3f3f3f3f333f3f3f3f3f3f3f3 f3H  0޽h ? ̙33___PPT10+ UD' *= @B D]' = @BA?%,( < +O%,( < +D' (=%(DG' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*b%(D' K=+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*bD' K=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*bD' (=%(DG' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*b%(D' K=+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*bD' K=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*bD' (=%(DG' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' K=+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*%D' K=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*%D' (=%(DG' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' K=+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*%D' K=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*%+8+0+ +g  p (    0<  ZProcedure Quicksort(A,l,r) Input :Unsorted Array (A,l,r) Output : Sorted subarray A(0..r) To sort the entire array A, l = 1 and r = length[A] if l < r then q PARTITION(A,l,r) QUICKSORT(A,l,q-1) QUICKSORT(A,q+1,r) >h 2    4  & 0HH8  P   P Z  s *  P Z  s *0  Z  s * 0    s *ȓ p   ;X 2   0ؘ p   ^ y X4 2   0 p `  d y X4 2H  0޽h ? ̙33      9 ( 0Wd >  0P0  Procedure PARTITION(A,l,r) Input : Array A(l .. r) Output : A and q such that A[i] A[q] for all i q and A[j] > A[q] for all j > q. x A[l]; i l; j r; while i < j do while A[i] x and i r do i i +1; while A[j] > x and j l do j j -1; if i< j then exchange A[i] A[j]; q j; exchange A[l] A[q];     %    F  P    `0Z  s *  P Z  s *0  Z  s * 0   s *8 p   ;X 2   0 p   ^ y X4 2   0 p `  d y X4 2H  0޽h ? ̙33  | (   l  C       0`   T(n) = n-1 + T(i-1) + T(n-i) It takes n-1 comparisons for the partition Then we sort smaller sequences of size i-1 and n-i Each element has the same probability of being selected as the pivot, The average running time is given by, Detailed Analysis in the Classx! 2`   c $A ??@   H  0޽h ? ̙33~  .&$(  $l $ C < @    $ 0 Like Quicksort, Mergesort algorithm also is based on the divide-and conquer principle. Divide: This step computes the middle of the array, takes constant time, Q(1) Conquer : Two subproblems, each of size n/2 are recursively solved. Each subproblem contributes 2T(n/2) to the running time. Combine: Two sorted sequences are merged, this takes Q(n) time o 2aA z . H $ 0޽h ? ̙33H  O G ( (  (l ( C #    3 ( 0X&@c,$  0 13 02 18 26 76 87 98 11 93 77 65 43 38 09 65 06 13 02 18 26 76 87 98 11 93 77 65 43 38 09 65 06 13 02 18 26 76 87 98 11 93 77 65 43 38 09 65 06 02 13 18 26 76 87 98 11 93 77 65 43 38 09 65 06 02 13 18 26 76 87 98 11 93 77 65 43 38 09 65 06 02 13 18 26 76 87 98 11 93 77 65 43 38 09 65 06 02 13 18 26 76 87 98 11 93 77 65 43 38 09 65 06 02 13 18 26 76 87 98 11 93 77 65 43 38 09 65 06 02 13 18 26 76 87 11 98 93 77 65 43 38 09 65 06 02 13 18 26 11 76 87 98 93 77 65 43 38 09 65 06 02 11 13 18 26 76 87 98 93 77 65 43 38 09 65 06 02 11 13 18 26 76 87 98 06 09 38 43 65 65 77 99 02 06 09 11 13 18 26 38 43 65 65 76 77 87 93 98  201G G3G3 3G33 3133 3933 3 3133 3 33133 3 3313 33,3 33,0303H ( 0޽h ? ̙33)<!<___PPT10<+}?PD;' *= @B Dh;' = @BA?%,( < +O%,( < +D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(a%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(aD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(aD{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(a%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(aD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(aD{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(#%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(#D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(#D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(#%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(#D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(#D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(F%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(FD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(FD{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(F%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(FD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(FD{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(i%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(iD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(iD{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(i%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(iD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(iD{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(+%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(+D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(+D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(+%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(+D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(+D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(+8+0+( +\   ,( v  ,d , 0~p3 ~Procedure MERGESORT(A,l,r) Input : A an array in the range 1 to n. Output: Sorted array A. if l < r then q (l+r )/2 ; MERGESORT(A,l,q) MERGESORT(A,q+1, r) MERGE (A,l,q,r) j[d 2D  H , 0޽h ? ̙33  R J 0 (  0 0 0  J Procedure MERGE(A,l,q,r) Inputs: two sorted subarrays A(l, q) and A(q+1, r) Output : Merged and sorted array A(l, r) i l; j q+1; k 0; while (i q) and ( j r) do k k+1; if A[i] A[j] then TEMP[k] A[i]; i i +1; else TEMP[k] A[j]; j j +1; if j > r then for t 0 to q  i do A[r-t] A[q-t]; for t 0 to k-1 do A[l +t] TEMP[t]; w 2     H 0 0޽h ? ̙33(  ''HL8B'(   8l 8 C 4    R 8 s *p 0: 8 3 o$ 8 6D =   8 6 T  @   8 6l  =n :  8 3 o}  8 6 - A "  8 6 An/2"  8 6Hr-Z H n :  8 3 o 8 6H] A " 8 6 ? An/2" 8 6H  =n  8 zBgCDEFgbg @`C  8 zBfCDEFbf @`1 : 8 3 od 8 6$ An/4" 8 6 )   An/2": 8 3 om 8 6.[ An/4" 8 6p2   An/2" 8 647> W =n  8 zBCDEFz  @` 8 zBECDEF ;E @`x: 8 3 o l 8 6X<  An/4" 8 6@   An/2"  8 6F> XW =n : !8 3 ou "8 6DK"h An/4" #8 6O "  An/2" $8 6U> "W =n  %8 zBCDEF{  @` &8 zBDCDEF ;D @`8 'Y  *8'Y B '8 3 'Y B (8 3 '- YZ B )8 3 ' Y 8 \  .8\ B +8 3 \ B ,8 3 \F r  -8 zB7CDEF272 @`\  @ /8 C o: 6k 08 6] W =   18 6b ! g  @   28 60f   =n @ 38 C o: k  48 6`k ;o  A " 58 6o  o  An/2"@ 78 C o. Z  88 6 uV Ujo  A " 98 6y Lc  An/2" ;8 zBkCDEFfkf @`_ U J  <8 zBfCDEFfb @`J C J @ =8 C oN v{  >8 6`   An/4" ?8 60. )G  An/2"@ @8 C oN { {  A8 6 'm  An/4" B8 6 . 'G  An/2" C8 zBCDEF{ { @`j :  D8 zB@CDEF @7 @`j B @ E8 C oN ~{  F8 6   An/4" G8 6. 1G  An/2"@ H8 C oN {  I8 6@ /u  An/4" J8 6l. /G  An/2" K8 zBCDEF{ { @`j :  L8 zB?CDEF ?6 @`j B H 8 0޽h ? ̙33  <Z(  <l < C      < 0,@ @ N.T(n) = 2 T(n/2) + Q(n) T(n/2) = 2 T(n/4) + n/2 T(n/4) = 2 T(n/8) + n/4 T(n) = 2{2 T(n/4) + n/2} + n = 4 T(n/4) + 2 n T(n) = 23 T(n/23) + 3n . . . T(n) = 2k T(n/2k) +k n If n = 2k then , k = log n Therefore, T(n) = 2k T(1) + n log n = n Q(1) + n log n T(n) = O(n log n)  2j     &  H < 0޽h ? ̙33  @(  @l @ C T    l @ C (p  H @ 0޽h ? ̙33j  D( v  Dr D S |  H D 0޽h ? ̙33$  |$(  |r | S lP   r | S ,  H | 0޽h ? ̙3380___PPT10.dq!P     T (  Tx T c $     T 0P05 RPorcedure PARALLEL ODD-EVEN MERGE SORT 1. for i 1 to N/2 ; N = number of data elements 2. for all PEj such that j is odd and 1 j N-1 3. if [PEj].a > [PEj+1].a 4. swap([PEj].a,[PEj+1].a) 5. for all PEj such that j is even and 1 j N-1 6. if [PEj].a > [PEj+1].a 7. swap([PEj].a,[PEj+1].a) } S 2 (    k T   @  51k T  3 52k T  3G6 53q T # Q @` 5nLB T c $D H T 0޽h ? ̙33V     P` (  `x ` c $p     ` 0D*@  ~A bitonic sequence is a sequence of numbers, a0,a1, . . ., an-1 such that for 0 i n-1 a0,a1, . . ., ai is in ascending order and ai+1,ai+2, . . ., an-1 is in descending order let, n = 2k, and i = n/2 -1. a0a1a2, . . ., an/2-2 an/2-1 . . . and an/2 an/2+1 an/2+2 an/2+3 . . . an-2 an-1 Consider, min (a0,an/2), min (a1,an/2+1), . . ., min(an/2-1,an-1) max(a0,an/2), max(a1,an/2+1), . . . , max(an/2-1,an-1) Both are bitonic sequences!! d/            %                            >`\H ` 0޽h ? ̙33  t%(  tx t c $Ti@    t 0j  5+$dB t <DodB t <DodB t <Do dB t <Do  t <o` 1a  t <s@` 9b  t <v`  @min(a,b)   t <}f  @max(a,b)   t 0J 5-$dB  t <Do dB t <Do dB t <DoJ*dB t <DoJ* t <`  1a t <P@ ` 9b t <8 @max(a,b)  t <Ԕf  @min(a,b) h t <  J JCompare a0 with an/2 a1 with an/2+1 . . . an/2-1 with a n-1 K  t <ܥ x ^Compare data elements i and 2k-1+i and put the min in position 2i and the max in position 2i+1*_? t < k5compare_exchange_min(j); compare_exchange_max(j); 6K6H t 0޽h ? ̙33X$  $$NNx#(  xx x c $Գp    x <  = 87 75 63 31 L @p x# @ N @p x @pN @  x @ Z x s *@Z x s *` @ N @   x @ @pZ  x s *@Z  x s *` @ lB  x <Do@@lB  x <Do x 0\pF M+  x 0Tp& M+  x 0p  M+  x 0 p  M+ dB x <Do pP dB x <DopPdB x <DopPdB x <DopPdB x <DopPdB x <Do&pP&dB x <Do pP dB x <Do pP L   x#  fB x 6Do  lB x <DofB xB 6Do L   x#  fB x 6Do  lB  x <DofB !xB 6Do L   "x#   fB #x 6Do  lB $x <DofB %xB 6Do L   &x# fB 'x 6Do  lB (x <DofB )xB 6Do L   *x# fB +x 6Do  lB ,x <DofB -xB 6Do L   .x#  fB /x 6Do  lB 0x <DofB 1xB 6Do L  @ 2x# &@ fB 3x 6DjJ `B 4x 0D p `B 5x 0D `B 6x 0D ` `B 7x 0D` P `B 8x 0D@ @`B 9x 0D p `B :x 0D P L  @ ;x# &p  fB x 0D `B ?x 0D ` `B @x 0D` P `B Ax 0D@ @`B Bx 0D p `B Cx 0D P  Dx <`0 + Ex < + Fx <L  + Gx <   + Hx 0 p F 1+ Ix 0L p & 1+ Jx 0 p   1+ Kx 0 p  1+ Lx <v  G31 45 63 67 75 78 87 89 Mx <R G 45 67 78 89 x  Nx < *  _#n/2 switches per stage log n stages $#H x 0޽h ? ̙33Zj  4 4`jjd3(  dF I @ d I @ d s *I \,$D 0 d s *@ S,$D 0 d s *6 I ,$D 0 d s *- @,$D 0 d 6܄f3,$D 0 + d 6f3H-,$D  0 +  d 6f3Ht - ,$D  0 +  d 6pf3Hl - ,$D  0 +Zz  I @  d I  @,$D 0  p  d#  I @,$D  0N @   d  Z d s *@Z d s *` @ N @  d @ pZ d s *@Z d s *` @  d 6  ,$D  0 + d 6|A |' ,$D 0 + d 6ȜA t ' ,$D 0 + d 6 ,$D 0 +Zz 0'' d 0'',$D 0  p d# 0'',$D 0N @  d  Z d s *@Z d s *` @ N @  d @ pZ d s *@Z d s *` @  d 6+,$D 0 +  d 6|k,$D 0 + !d 6t k ,$D 0 + "d 6|l k ,$D 0 +"z G #d G,$D 0H   @ $d# G,$D 0fB %d 6Do  fB &d 6Do  fB 'd 6Do  fB (d 6Do @ @H   @ )d# Gt ,$D 0fB *d 6Do  fB +d 6Do  fB ,d 6Do  fB -d 6Do @ @ .d 0W,$D 0 U24 42 83 98 93 68 28 04"    /d# ,$D 0T   @ 0d#   @fB 1d 6Do  fB 2d 6Do  fB 3d 6Do  fB 4d 6Do @ @T   @ 5d#  `  fB 6d 6Do  fB 7d 6Do  fB 8d 6Do  fB 9d 6Do @ @`B :d 0D  `B ;d 0D  ` `B d 0D  `B ?d 0D `B @d 0D @ 0 `B Ad 0D     Bd#   ,$D 0T   @ Cd#   @fB Dd 6Do  fB Ed 6Do  fB Fd 6Do  fB Gd 6Do @ @T   @ Hd#  `  fB Id 6Do  fB Jd 6Do  fB Kd 6Do  fB Ld 6Do @ @`B Md 0D  `B Nd 0D  ` `B Od 0D  `B Pd 0D @ `B Qd 0D  `B Rd 0D `B Sd 0D @ 0 `B Td 0D     Ud#  ,$D 0T   @ Vd#   @fB Wd 6Do  fB Xd 6Do  fB Yd 6Do  fB Zd 6Do @ @T   @ [d#  `  fB \d 6Do  fB ]d 6Do  fB ^d 6Do  fB _d 6Do @ @`B `d 0D  `B ad 0D  ` `B bd 0D  `B cd 0D @ `B dd 0D  `B ed 0D `B fd 0D @ 0 `B gd 0D  hd 0ɝ x ,$D 0 U24 93 42 68 28 83 04 98" id 0Ν W,$ 0 U24 28 83 93 04 42 68 98" jd 04ӝG,$D  0 U04 24 28 42 68 83 93 98"H d 0޽h ? 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Oh+'0 hp   @ L XdlQuicksort and Mergesort Mohan KumarQC:\Program Files\Microsoft Office\Templates\Presentations\Generic (Standard).potr Mohan Kumar10aMicrosoft PowerPointoso@jN@v-L@pb[WG g  <1  -- @ !--'@Times New Roman-. 2 U 8/31/2004b   ."System-@Times New Roman-. 12 ~M KUMAR CSE5311  .-@Times New Roman-.  2 1 .-@Times New Roman-. *2 O#Quicksort and Mergesort#   *.-@Times New Roman-. 3332 < This week  .-@Wingdings-. 333 2 *q.-@Arial-. 333$2 *Quicksort algorithms    .-@Wingdings-. 333 2 Zq.-@Arial-. 333'2 ZQuicksort performancer   .-@Wingdings-. 333 2 q.-@Arial-. 333"2 Quicksort analysis    .-@Wingdings-. 333 2 q.-@Arial-. 333$2 Mergesort algorithmc    .-@Wingdings-. 333 2 q.-@Arial-. 333"2 Mergesort analysis    .-@Arial-. 2 Further Reading   .-@Arial-. 33"2 Chapters 1 and 8      .-@Arial-. 332 , from Textbooki   .-՜.+,0$     On-screen Show Curtin University,A Times New RomanArial WingdingsSymbolDefault DesignMicrosoft Equation 3.0Quicksort and Mergesort QuicksortExampleSlide 4Slide 5Running Time of Quicksort MergesortExampleSlide 9 Slide 10 Runtime Complexity of Mergesort Runtime Complexity of Mergesort Questions Slide 14 Slide 15The Odd-Even Mergesort problemC 9 18 27 36 45 54 63 72 64 56 48 40 32 24 16 8 Bitonic MergesortSorting Networks "Shuffle-Exchange Network Week 6 Slide 20!Parallel Bitonic Sort Algorithm Given Unsorted list : Slide 23  Fonts UsedDesign TemplateEmbedded OLE Servers Slide Titles#_a Mohan KumarMohan Kumar  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&()*+,-.3Root EntrydO)PicturesCurrent User'SummaryInformation(PowerPoint Document(DocumentSummaryInformation8cse5311-old/Module2_Additional.pdf010064400020320000200000020451331011613146500173020ustar00cse5311cse531100000400000024%PDF-1.3 % 884 0 obj << /Linearized 1 /O 886 /H [ 2208 1391 ] /L 543323 /E 152400 /N 36 /T 525524 >> endobj xref 884 88 0000000016 00000 n 0000002111 00000 n 0000003599 00000 n 0000003819 00000 n 0000004648 00000 n 0000004700 00000 n 0000004752 00000 n 0000004804 00000 n 0000004845 00000 n 0000004897 00000 n 0000005459 00000 n 0000005895 00000 n 0000006102 00000 n 0000006328 00000 n 0000006555 00000 n 0000007063 00000 n 0000007086 00000 n 0000011521 00000 n 0000011544 00000 n 0000015659 00000 n 0000015682 00000 n 0000019936 00000 n 0000019959 00000 n 0000024147 00000 n 0000024170 00000 n 0000028504 00000 n 0000028527 00000 n 0000032870 00000 n 0000033104 00000 n 0000033755 00000 n 0000033900 00000 n 0000033923 00000 n 0000038314 00000 n 0000038732 00000 n 0000039125 00000 n 0000040125 00000 n 0000040774 00000 n 0000040797 00000 n 0000045661 00000 n 0000064789 00000 n 0000083719 00000 n 0000083798 00000 n 0000106898 00000 n 0000107704 00000 n 0000107955 00000 n 0000108478 00000 n 0000111156 00000 n 0000111370 00000 n 0000137093 00000 n 0000137741 00000 n 0000144099 00000 n 0000144743 00000 n 0000144959 00000 n 0000145173 00000 n 0000145387 00000 n 0000145603 00000 n 0000145815 00000 n 0000146031 00000 n 0000146240 00000 n 0000146454 00000 n 0000146668 00000 n 0000146880 00000 n 0000147089 00000 n 0000147306 00000 n 0000147519 00000 n 0000147734 00000 n 0000147948 00000 n 0000148160 00000 n 0000148377 00000 n 0000148593 00000 n 0000148810 00000 n 0000149019 00000 n 0000149232 00000 n 0000149447 00000 n 0000149664 00000 n 0000149880 00000 n 0000150097 00000 n 0000150311 00000 n 0000150520 00000 n 0000150739 00000 n 0000150956 00000 n 0000151167 00000 n 0000151382 00000 n 0000151595 00000 n 0000151813 00000 n 0000152030 00000 n 0000002208 00000 n 0000003576 00000 n trailer << /Size 972 /Info 882 0 R /Root 885 0 R /Prev 525513 /ID[<88d843683fd645175eb029ee2cef586b>] >> startxref 0 %%EOF 885 0 obj << /Type /Catalog /Pages 878 0 R /Metadata 883 0 R /PageLabels 876 0 R >> endobj 970 0 obj << /S 1453 /L 1694 /Filter /FlateDecode /Length 971 0 R >> stream HT}LSWﳏ~~ {>Mڨ1ag_k̴؇~n1[WJ+WsP+Ӕ>3˃ߝKdZRI̢)rV#k@iʜQ_*2 eJ˳3=mY`O7^RvZX%S {$\Y/mm :u#z-v?"2:hdO(; 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1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H1cF||<@ ( endstream endobj 90 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 31 /Filter /FlateDecode >> stream H:u… ߽{w9k M endstream endobj 91 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream HZhQFFFfffvvׯ =\ endstream endobj 92 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 32 /Filter /FlateDecode >> stream Hڶm[FFFff&|͛srr endstream endobj 93 0 obj << /Type /XObject /Subtype /Image /Width 11 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 45 /Filter /FlateDecode >> stream H#CMMɓ˗/O>]GGٳgBk endstream endobj 94 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hz𡃃 K endstream endobj 95 0 obj << 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101 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HZ|9CII @iR endstream endobj 102 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HZd@  endstream endobj 103 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream Hڱcׯ_?ndd`Y endstream endobj 104 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream H:sӧO]`> stream Hڽ{wyyyQQQqq)S B endstream endobj 106 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream Hjii),,0 h endstream endobj 107 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HRQQ'@  endstream endobj 108 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H:}4˗/  endstream endobj 109 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 39 /Filter /FlateDecode >> stream Hzׯ=z$)) &&(** ` endstream endobj 110 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hz͛7 Y endstream endobj 111 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HZlY8*> stream H5kÒ%K Q endstream endobj 113 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Huí[  endstream endobj 114 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 42 /Filter /FlateDecode >> stream HਨCCC˝ a endstream endobj 115 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hwӧ  endstream endobj 116 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hz9Cnn.@? endstream endobj 117 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H*..z9@- endstream endobj 118 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H:póg m endstream endobj 119 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HZr@  endstream endobj 120 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hٱc@ endstream endobj 121 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hzӧ [ endstream endobj 122 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HZti*8q"@`K endstream endobj 123 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream H[bbbPPЁ 2(< endstream endobj 124 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream H׎;---= `I endstream endobj 125 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HpႬǏ q endstream endobj 126 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream HVTT??eSS@? endstream endobj 127 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hڹs͛7  endstream endobj 128 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 20 /Filter /FlateDecode >> stream HQAEEE: endstream endobj 129 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H03 endstream endobj 130 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream H[uuuyyyeeٳ N' endstream endobj 131 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H*//}6@ ! endstream endobj 132 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 31 /Filter /FlateDecode >> stream Hڲe˴i~U]]]YYys endstream endobj 133 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H211ILL0X endstream endobj 134 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hc``7o@U endstream endobj 135 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:s͛7  endstream endobj 136 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H  endstream endobj 137 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hzc  endstream endobj 138 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hˋ'@uD endstream endobj 139 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hω' U" endstream endobj 140 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hu+@^ endstream endobj 141 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream HzqHHڵk ,^ endstream endobj 142 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 31 /Filter /FlateDecode >> stream Hrrr 6660O endstream endobj 143 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H}}  endstream endobj 144 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HsNVV@f endstream endobj 145 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HZ`@ endstream endobj 146 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hr努˗/  endstream endobj 147 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream Hz5k<<<> `8aw endstream endobj 148 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HzU A endstream endobj 149 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H;w-@  endstream endobj 150 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 33 /Filter /FlateDecode >> stream Hrvvnjjڹs'͛7 c I endstream endobj 151 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 23 /Filter /FlateDecode >> stream H rssKLL0 endstream endobj 152 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H,//0q endstream endobj 153 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 34 /Filter /FlateDecode >> stream H>{ׯ988&L ##er  endstream endobj 154 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HΞ> stream H:<@n endstream endobj 156 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hŋ? `& endstream endobj 157 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream Hڸq=ûw "Y endstream endobj 158 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H{7   endstream endobj 159 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hr  endstream endobj 160 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream H:~8 ߯^`G I endstream endobj 161 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream H􌉉 ˲e˦M`K  endstream endobj 162 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H*))y%@ H endstream endobj 163 0 obj << /Type /XObject /Subtype /Image /Width 1 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 15 /Filter /FlateDecode >> stream H@ endstream endobj 164 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 35 /Filter /FlateDecode >> stream Hz_|$ 3 endstream endobj 165 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H0 endstream endobj 166 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HC||<@P endstream endobj 167 0 obj << /Type /XObject /Subtype /Image /Width 1 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 15 /Filter /FlateDecode >> stream H'@ endstream endobj 168 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream HGppU / endstream endobj 169 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hڿ?;;w  endstream endobj 170 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H277/))0 endstream endobj 171 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 33 /Filter /FlateDecode >> stream H庺 { endstream endobj 172 0 obj << /Type /XObject /Subtype /Image /Width 11 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 48 /Filter /FlateDecode >> stream H!^^^iii&&&''';;;[[[  endstream endobj 173 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H|g %\ endstream endobj 174 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream H~Ǐ=<< 8 endstream endobj 175 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H{npp۷o ,X endstream endobj 176 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HSff&@ endstream endobj 177 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream H:y򤪪ѣG ?r endstream endobj 178 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream HKllUff&@N endstream endobj 179 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream HZzņ ^ `*9 endstream endobj 180 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H@  endstream endobj 181 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H0 endstream endobj 182 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hڰa@ ) endstream endobj 183 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream H:qℎCRRRqq*@? endstream endobj 184 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 15 /Filter /FlateDecode >> stream H \ endstream endobj 185 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H:wÌ3 3 endstream endobj 186 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H5k@ endstream endobj 187 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hgnn.@+3 endstream endobj 188 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream HUTTd``X~=''m -6 endstream endobj 189 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream Hڲe.CIIIRR@=2 endstream endobj 190 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hfbbڵk@ F endstream endobj 191 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HZdҭ[  endstream endobj 192 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H󧔔Tss3@ endstream endobj 193 0 obj << /Type /XObject /Subtype /Image /Width 13 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 49 /Filter /FlateDecode >> stream HڰaǷo^~}񰰰?vtt& endstream endobj 194 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream HrvvNOO0] endstream endobj 195 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H5k@ endstream endobj 196 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 32 /Filter /FlateDecode >> stream HgNNNLLLXXZ \~ e endstream endobj 197 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H}v]]̙3  endstream endobj 198 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hbff^l@ endstream endobj 199 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hz#"" " endstream endobj 200 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H*,,@> endstream endobj 201 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream Hzݻw (D endstream endobj 202 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 28 /Filter /FlateDecode >> stream H:pݻw-Zt)F | endstream endobj 203 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H={6//Ç B endstream endobj 204 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HZz5##ϟ  endstream endobj 205 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H0# endstream endobj 206 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream HJIIݾ}ի~ `I endstream endobj 207 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H[JJJdd$@  endstream endobj 208 0 obj << /Type /XObject /Subtype /Image /Width 1 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 14 /Filter /FlateDecode >> stream H0~ endstream endobj 209 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H> stream Hw۷ϟ?u }`o endstream endobj 211 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 34 /Filter /FlateDecode >> stream Hz//;w.\ ''/n  endstream endobj 212 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream Hgxx  P: endstream endobj 213 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream H{VTTݻw`D endstream endobj 214 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:r+++@ \ endstream endobj 215 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HqƎ;  endstream endobj 216 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H}w % endstream endobj 217 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 42 /Filter /FlateDecode >> stream H񑓓+//722sŋ yxxjjj> ` endstream endobj 218 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 36 /Filter /FlateDecode >> stream H wvv~ u/_ `z+ x endstream endobj 219 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream H>CRRRfffll,@6* endstream endobj 220 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HZ~ŋ  endstream endobj 221 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HGNN@- endstream endobj 222 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 33 /Filter /FlateDecode >> stream Hwý{`r { endstream endobj 223 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream HrvvzҥK @D endstream endobj 224 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream H:vXvvjGG@( endstream endobj 225 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 32 /Filter /FlateDecode >> stream HڹsCzz˗/ld_ endstream endobj 226 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream Hʼ _O : endstream endobj 227 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 25 /Filter /FlateDecode >> stream Hzwq? m endstream endobj 228 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HJKK|2@  endstream endobj 229 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H۷ 9/ endstream endobj 230 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 28 /Filter /FlateDecode >> stream H:{Ç߿_UU}v\  endstream endobj 231 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream Hν eC R endstream endobj 232 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream Hsss0 endstream endobj 233 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HCll,@D endstream endobj 234 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream Hs~Ǐ endstream endobj 235 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 33 /Filter /FlateDecode >> stream H  endstream endobj 236 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream H훃#G ,gX endstream endobj 237 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 36 /Filter /FlateDecode >> stream Hݾ͹𭭭 4 endstream endobj 238 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H:uCPP@U endstream endobj 239 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hjll{@C endstream endobj 240 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hz⅞իW y endstream endobj 241 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hzc   endstream endobj 242 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HۢE < endstream endobj 243 0 obj << /Type /XObject /Subtype /Image /Width 1 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 15 /Filter /FlateDecode >> stream H @ endstream endobj 244 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:|0;;;@ _ endstream endobj 245 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hz'@] endstream endobj 246 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hܾ};@ 1 endstream endobj 247 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hyf}}=@ endstream endobj 248 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H:qD@@ׯ_ 7( endstream endobj 249 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:vŋ r< endstream endobj 250 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H@X endstream endobj 251 0 obj << /Type /XObject /Subtype /Image /Width 1 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 15 /Filter /FlateDecode >> stream H@ endstream endobj 252 0 obj << /Type /XObject /Subtype /Image /Width 1 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 15 /Filter /FlateDecode >> stream H@ endstream endobj 253 0 obj << /Type /XObject /Subtype /Image /Width 1 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 15 /Filter /FlateDecode >> stream H;@ endstream endobj 254 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hڼy3CRR@@ endstream endobj 255 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hy󦪪j}}=@x endstream endobj 256 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream HZbGRR@BE^ endstream endobj 257 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Ht钹  endstream endobj 258 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HZ|y\\@  endstream endobj 259 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H}@0 endstream endobj 260 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream HZjGZZ@Bs endstream endobj 261 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H4i@ endstream endobj 262 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 42 /Filter /FlateDecode >> stream H<<> stream HOOO322ڼy@M endstream endobj 264 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream HJLL0  endstream endobj 265 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HK\\ܫW ! endstream endobj 266 0 obj << /Type /XObject /Subtype /Image /Width 13 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 49 /Filter /FlateDecode >> stream H CCÇfggwuumܸԩSvvv_|0[! endstream endobj 267 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream H:tƍݻ `k endstream endobj 268 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HjjjZr%@  endstream endobj 269 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HӧO  endstream endobj 270 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream Hz}GGׯ_/_nbb`Z endstream endobj 271 0 obj << /Type /XObject /Subtype /Image /Width 1 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 14 /Filter /FlateDecode >> stream H0u endstream endobj 272 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hjkk{=@ z` endstream endobj 273 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HRQQYz5@$n endstream endobj 274 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 37 /Filter /FlateDecode >> stream H:ujCCCdd}pBsss F endstream endobj 275 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hܶm۲e o endstream endobj 276 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream H***ڴi͛7%$$?~,-- ``$ : endstream endobj 277 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream Hwww0 !S endstream endobj 278 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:zI  endstream endobj 279 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H[ _h endstream endobj 280 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hruu1c@ endstream endobj 281 0 obj << /Type /XObject /Subtype /Image /Width 1 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 15 /Filter /FlateDecode >> stream Hu@ endstream endobj 282 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 33 /Filter /FlateDecode >> stream Hjll|w<<<|}}{zz  t endstream endobj 283 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hjkk[b@  endstream endobj 284 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hz+$7t endstream endobj 285 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream H:ySSS=z`Qi endstream endobj 286 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hrvv3g@ endstream endobj 287 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H{m,X`#+ endstream endobj 288 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream Hze``o?@?W endstream endobj 289 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 42 /Filter /FlateDecode >> stream H...ppp  endstream endobj 290 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:~ 9 endstream endobj 291 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream Hڶm߱c 7 endstream endobj 292 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H>y$@Q endstream endobj 293 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:w6@  endstream endobj 294 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 26 /Filter /FlateDecode >> stream Hjkkc`` sqq0- endstream endobj 295 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hںu5@  endstream endobj 296 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 20 /Filter /FlateDecode >> stream H*))E6m0= endstream endobj 297 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H;;;{oo/@ endstream endobj 298 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H>{,@i endstream endobj 299 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H> stream Hg`` 01 endstream endobj 301 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HzMzzǏ ! endstream endobj 302 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H*//@f endstream endobj 303 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hսx"@ endstream endobj 304 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H~ŋ qo endstream endobj 305 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream H뗅9s H endstream endobj 306 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HRUUݿ?@ endstream endobj 307 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:v옜@ T endstream endobj 308 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream H:~x~~+fΜ`IwD endstream endobj 309 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HZz&@ } endstream endobj 310 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HZ|9@  endstream endobj 311 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HRQQ9|0@ endstream endobj 312 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H4i@b endstream endobj 313 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream H:zh^^ުUN`IA endstream endobj 314 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HZn@ ( endstream endobj 315 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H󧱱5k f endstream endobj 316 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H㇌ɓ'  endstream endobj 317 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 29 /Filter /FlateDecode >> stream HZ|yKKKzzqW m endstream endobj 318 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 31 /Filter /FlateDecode >> stream H?~lool2v _ endstream endobj 319 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H:|0??s  endstream endobj 320 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:v올,@ N endstream endobj 321 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 39 /Filter /FlateDecode >> stream HJJJ깹((( ; endstream endobj 322 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 20 /Filter /FlateDecode >> stream HF[l0A(i endstream endobj 323 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream Hz葡={v `1, endstream endobj 324 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H:p??K  endstream endobj 325 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H8q"@\ endstream endobj 326 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H@ endstream endobj 327 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream HZ endstream endobj 328 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream H<{ +n endstream endobj 329 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H=y$@o endstream endobj 330 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 29 /Filter /FlateDecode >> stream HZ~  BCCcǎ^  endstream endobj 331 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hڷo_ll,@ VR endstream endobj 332 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H 04 endstream endobj 333 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream Hzy||K -< endstream endobj 334 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HpႾ>@ P endstream endobj 335 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 37 /Filter /FlateDecode >> stream Hz9۝;wLMM/_b {{{z endstream endobj 336 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 20 /Filter /FlateDecode >> stream HrqqDӧO0^ endstream endobj 337 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 36 /Filter /FlateDecode >> stream H ߹sgmmׯeee/_ ` . endstream endobj 338 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H2e @ endstream endobj 339 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 37 /Filter /FlateDecode >> stream H>ݻw-,,N:emm}Q; endstream endobj 340 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 39 /Filter /FlateDecode >> stream Hiii&&& J endstream endobj 341 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HvڢE \i endstream endobj 342 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream Hjnn:r䈤dpp I endstream endobj 343 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream Hھ}{DDDtttoo/@( endstream endobj 344 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 41 /Filter /FlateDecode >> stream HjkkǏsss={ d.Y : endstream endobj 345 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 35 /Filter /FlateDecode >> stream H0aBlllttm>dO0 endstream endobj 346 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hڵkʕ+ * endstream endobj 347 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 33 /Filter /FlateDecode >> stream Hzmnn322bcc߿ ` Q endstream endobj 348 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream HzaXXX|||rr2@- endstream endobj 349 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream Hjlltqq0 -P endstream endobj 350 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:~i 3 endstream endobj 351 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream Hھ}{ww͛7ë LS endstream endobj 352 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H_f @ " endstream endobj 353 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H {.@ M endstream endobj 354 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:x`PP@ A: endstream endobj 355 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:zhpp0@ I endstream endobj 356 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H2e @ X endstream endobj 357 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:z)S   endstream endobj 358 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HڶmΧO Y} endstream endobj 359 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HZfMPP@  endstream endobj 360 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H̙3 \ endstream endobj 361 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:uI -v endstream endobj 362 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HჵÇ D% endstream endobj 363 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H˗/  endstream endobj 364 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H?W  endstream endobj 365 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hڵk rr endstream endobj 366 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hx _ endstream endobj 367 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HڷoߥK  endstream endobj 368 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hz g endstream endobj 369 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H7 t endstream endobj 370 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 33 /Filter /FlateDecode >> stream H  endstream endobj 371 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hڹs̙3 s endstream endobj 372 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H+@ endstream endobj 373 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 28 /Filter /FlateDecode >> stream Hz/^zU]]Of0 | endstream endobj 374 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hu|`&R endstream endobj 375 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hz @7 endstream endobj 376 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 39 /Filter /FlateDecode >> stream H씒upp066~޺u s; endstream endobj 377 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hzqww? #F3 endstream endobj 378 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H+,,0.n endstream endobj 379 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H}6@ `j endstream endobj 380 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream Hv횩)77} %} endstream endobj 381 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H0a;;;@ endstream endobj 382 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 37 /Filter /FlateDecode >> stream Hزeڵk>|k׮) endstream endobj 383 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HZnի : endstream endobj 384 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hzꕜ\EE@ endstream endobj 385 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream Hꪮ0  endstream endobj 386 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H{ s endstream endobj 387 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 23 /Filter /FlateDecode >> stream Hjoowqq퍍0"D endstream endobj 388 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream Hz򥄄۷.\`; endstream endobj 389 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream Hjmm0 w endstream endobj 390 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HZp}V^ `E endstream endobj 391 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 25 /Filter /FlateDecode >> stream Hzѽ{nܸD. endstream endobj 392 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hrvv;w.@ endstream endobj 393 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 20 /Filter /FlateDecode >> stream Hjiia``hnn0{ endstream endobj 394 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream Hjoo 0  endstream endobj 395 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:|p?@ endstream endobj 396 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream H?~JJٳg * endstream endobj 397 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 25 /Filter /FlateDecode >> stream Hz;wnݺGC endstream endobj 398 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HZ~ $p%k endstream endobj 399 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream HZhQ||W &O endstream endobj 400 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HRQQ9x @ endstream endobj 401 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H> stream H:pR||<@%_ endstream endobj 403 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream H>p222\\\ ] X endstream endobj 404 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HzUrrw !Z endstream endobj 405 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:qČ3 N! endstream endobj 406 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 33 /Filter /FlateDecode >> stream Hjnn644\r%|]]@J+ endstream endobj 407 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H7o^38y$@^ Z endstream endobj 408 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream HJOO5k۷o #C endstream endobj 409 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HRTT> stream H2558q"@S endstream endobj 411 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 26 /Filter /FlateDecode >> stream HjhhPQQlnn0*8 endstream endobj 412 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hxbhh(@Cs endstream endobj 413 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream H:|paaarrrQQ@,@ endstream endobj 414 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H> stream H:y3 ]$ endstream endobj 416 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 32 /Filter /FlateDecode >> stream HRTT)//HOO0JH endstream endobj 417 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 40 /Filter /FlateDecode >> stream HzG``Ǐ.--}! ~ endstream endobj 418 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H4i@n endstream endobj 419 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream HzUUU۷Onkk `T : endstream endobj 420 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H1c @ &e endstream endobj 421 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream HsssSTTTRRZt)@& endstream endobj 422 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H?}4@ endstream endobj 423 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 28 /Filter /FlateDecode >> stream Hjll2.] `; endstream endobj 424 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 37 /Filter /FlateDecode >> stream HZj_z˗h endstream endobj 425 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 20 /Filter /FlateDecode >> stream H244d@ #; endstream endobj 426 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 28 /Filter /FlateDecode >> stream H˗/ϝ;'%%#a>  endstream endobj 427 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HZh˗/ 5 endstream endobj 428 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H2228q"@ZJ endstream endobj 429 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 37 /Filter /FlateDecode >> stream Hzꕚڷo괵o޼iffh" ! endstream endobj 430 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream HëWBBBrrr~`Y endstream endobj 431 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H>~8@E endstream endobj 432 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 29 /Filter /FlateDecode >> stream HzգG6nd|  endstream endobj 433 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 42 /Filter /FlateDecode >> stream Hz𡱱7-[v}}?&$$ϟ?0T endstream endobj 434 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H2eJ8r@V endstream endobj 435 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 33 /Filter /FlateDecode >> stream HZh##ccc-[%%%-Z`\S endstream endobj 436 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H={6@ C endstream endobj 437 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 37 /Filter /FlateDecode >> stream Hz𡠠ץ N>p- I endstream endobj 438 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 34 /Filter /FlateDecode >> stream Hޑ#G?nll| endstream endobj 439 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:sٳ 6 endstream endobj 440 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 43 /Filter /FlateDecode >> stream H0a@uuull_iee endstream endobj 441 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 36 /Filter /FlateDecode >> stream H˴̫ h@ endstream endobj 442 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:r̙3  endstream endobj 443 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 36 /Filter /FlateDecode >> stream H䀀 tr endstream endobj 444 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 31 /Filter /FlateDecode >> stream Hƍ׮]{ڵp  endstream endobj 445 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:rĉ  endstream endobj 446 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 42 /Filter /FlateDecode >> stream H沲 *. endstream endobj 447 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HpႎΣG i endstream endobj 448 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HZbEDD@  endstream endobj 449 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H*((ػw/@  endstream endobj 450 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H|ڵk q endstream endobj 451 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H*))inn0  endstream endobj 452 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H={͛ M endstream endobj 453 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hzⅽׯ 7 endstream endobj 454 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H;w t endstream endobj 455 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H S endstream endobj 456 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hݻw qq endstream endobj 457 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hz * endstream endobj 458 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hڽ{_  endstream endobj 459 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HzݡC  endstream endobj 460 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hs玓'@=6 endstream endobj 461 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream HZjGzz:@C#| endstream endobj 462 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H0s endstream endobj 463 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 42 /Filter /FlateDecode >> stream Hyyy===777 B endstream endobj 464 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream HGJJ˥K 2LW endstream endobj 465 0 obj << /Type /XObject /Subtype /Image /Width 13 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 52 /Filter /FlateDecode >> stream H700xQKKKVVVgg N:egggZ endstream endobj 466 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream H:tƍݻ`l endstream endobj 467 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HjjjZj@  endstream endobj 468 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hg  endstream endobj 469 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream H={fWW@ endstream endobj 470 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 37 /Filter /FlateDecode >> stream HrqqYdIsss``}h"sss  endstream endobj 471 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hضm۲e W endstream endobj 472 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream H*..޴iӭ[޿/** `bX | endstream endobj 473 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 23 /Filter /FlateDecode >> stream Hrpp05s endstream endobj 474 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:zɓ  endstream endobj 475 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H$w endstream endobj 476 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream HJJJ255ݳgσ * endstream endobj 477 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hrqq5k@ endstream endobj 478 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H},X`#  endstream endobj 479 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream HzuhhO@@ĉ @H endstream endobj 480 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream Hݰaóg Kc{ endstream endobj 481 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:qDAA@^ endstream endobj 482 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 20 /Filter /FlateDecode >> stream H*..E7n0=k endstream endobj 483 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 42 /Filter /FlateDecode >> stream H޶...MMMLLLQQQ _ endstream endobj 484 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 20 /Filter /FlateDecode >> stream Hڸq#< G endstream endobj 485 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:{,@Z endstream endobj 486 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H> stream Hjhh``` 0  endstream endobj 488 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 33 /Filter /FlateDecode >> stream HzmzzO|PXX`< endstream endobj 489 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H׿x"@ endstream endobj 490 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream H\ljnn.@! endstream endobj 491 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 36 /Filter /FlateDecode >> stream Hbbbڶm[AAcFF@aW endstream endobj 492 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream H:|pNNΊ+N`HN2 endstream endobj 493 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H4i@_ endstream endobj 494 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HZ~@ 7 endstream endobj 495 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream H:ydjjٳ>|`BT2 endstream endobj 496 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 33 /Filter /FlateDecode >> stream H9|pBB򊊊 f endstream endobj 497 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 29 /Filter /FlateDecode >> stream HZre{{{VVV@@q!Y endstream endobj 498 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 20 /Filter /FlateDecode >> stream H G۶m0C endstream endobj 499 0 obj << /Type /XObject /Subtype /Image /Width 1 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 14 /Filter /FlateDecode >> stream Hjmm0! endstream endobj 500 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HڵkWYY@  endstream endobj 501 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H:x ??K  endstream endobj 502 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream H2eccckjj :z] endstream endobj 503 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H@) endstream endobj 504 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:x+ E< endstream endobj 505 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H!@  endstream endobj 506 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 31 /Filter /FlateDecode >> stream HڵkԩSO8s???k o endstream endobj 507 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 29 /Filter /FlateDecode >> stream HZvmOOOnnnpp0qQ\( endstream endobj 508 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H@Q endstream endobj 509 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 20 /Filter /FlateDecode >> stream HrvvGӦM0@ endstream endobj 510 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 36 /Filter /FlateDecode >> stream H0aBUUUKK˗/cbbdee\`( 7 endstream endobj 511 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H2e @ endstream endobj 512 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 37 /Filter /FlateDecode >> stream H>ݻwMMM-,,N:emm} endstream endobj 513 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 39 /Filter /FlateDecode >> stream H...???%%% Y endstream endobj 514 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hvŋ bl endstream endobj 515 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 43 /Filter /FlateDecode >> stream Hjkk311Y`ױcDEEܹlggw]P endstream endobj 516 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 20 /Filter /FlateDecode >> stream Hjnnf``0 Y endstream endobj 517 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream HڹsgXXXTTT?@( endstream endobj 518 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hڹsʕ+ ' endstream endobj 519 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 33 /Filter /FlateDecode >> stream HzMvvgcbb߿ ` ? endstream endobj 520 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream Hz1âE "; endstream endobj 521 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream Hsvv0D endstream endobj 522 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HG__߅  ${ endstream endobj 523 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 45 /Filter /FlateDecode >> stream HrrrVVVccc  endstream endobj 524 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:x`pp0@ G= endstream endobj 525 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H> stream Hڻw  y endstream endobj 527 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HͬY \ endstream endobj 528 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H?U S endstream endobj 529 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:uĉ 's endstream endobj 530 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H񣍍Ç k+ endstream endobj 531 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 3 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 33 /Filter /FlateDecode >> stream H䢢 uf p endstream endobj 532 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hxu Y~ endstream endobj 533 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hzy a endstream endobj 534 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 2 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream Hڿ;wZ[[ + endstream endobj 535 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 3 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 31 /Filter /FlateDecode >> stream HW+%%u~I y endstream endobj 536 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HڹsgRRҳg  endstream endobj 537 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream Hjjj0 z endstream endobj 538 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HڻwoHH@ 4 endstream endobj 539 0 obj << /Type /XObject /Subtype /Image /Width 1 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 15 /Filter /FlateDecode >> stream H3@ endstream endobj 540 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hϟ?  endstream endobj 541 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H0 M endstream endobj 542 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 45 /Filter /FlateDecode >> stream H___ 28 endstream endobj 543 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HZt  endstream endobj 544 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:qDnn.@L endstream endobj 545 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 28 /Filter /FlateDecode >> stream Hw޳gϖ.]ziW A endstream endobj 546 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HYp  endstream endobj 547 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H߷o@  endstream endobj 548 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H%KΝ;`%B endstream endobj 549 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 20 /Filter /FlateDecode >> stream HSRR0> endstream endobj 550 0 obj << /Type /XObject /Subtype /Image /Width 13 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 49 /Filter /FlateDecode >> stream HTPPqƞ={899dJJ P͛7f0/ endstream endobj 551 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream Hze?"## q endstream endobj 552 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 20 /Filter /FlateDecode >> stream HRWWWVV0 endstream endobj 553 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:ujcc#@ C endstream endobj 554 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H㇉@)o endstream endobj 555 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 51 /Filter /FlateDecode >> stream H$666iii:::!!! 1 endstream endobj 556 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H*..޳gO~~>@o endstream endobj 557 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:rɓ  endstream endobj 558 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 33 /Filter /FlateDecode >> stream H߿۷ Yg endstream endobj 559 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hz葑Ǐ  endstream endobj 560 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H3gί_  endstream endobj 561 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hj~~>@ endstream endobj 562 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 39 /Filter /FlateDecode >> stream Hھ}{}}}jjjooohh(ӧW^ `{ H endstream endobj 563 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HZh=@  endstream endobj 564 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HںuӧO` pF endstream endobj 565 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream H𼼼 ` endstream endobj 566 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H0Y endstream endobj 567 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 33 /Filter /FlateDecode >> stream HzYJJJ on endstream endobj 568 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hz⅙ ( endstream endobj 569 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H@ Q endstream endobj 570 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream HzÇw`0 endstream endobj 571 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 25 /Filter /FlateDecode >> stream H|={6mt1;  endstream endobj 572 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H4iR 8|0@Ub p endstream endobj 573 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 34 /Filter /FlateDecode >> stream H뗡a||9svYUUS endstream endobj 574 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H2331c@k endstream endobj 575 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HZlYYY@ U endstream endobj 576 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream H g``^b@Zv endstream endobj 577 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H*..^t)@ I endstream endobj 578 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 26 /Filter /FlateDecode >> stream Ht钨hb00[ endstream endobj 579 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H^p!@ U endstream endobj 580 0 obj << /Type /XObject /Subtype /Image /Width 1 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 14 /Filter /FlateDecode >> stream Hjjj0 endstream endobj 581 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 37 /Filter /FlateDecode >> stream H񣚚!òe˄ uuu @ endstream endobj 582 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H:pz$pyh endstream endobj 583 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hu77+W , endstream endobj 584 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hrpp4i@2w endstream endobj 585 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 15 /Filter /FlateDecode >> stream HJHA endstream endobj 586 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream HX~ˊ+ . endstream endobj 587 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hz򥕕՗/_ F endstream endobj 588 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H7660 endstream endobj 589 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H02 endstream endobj 590 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 33 /Filter /FlateDecode >> stream Hsss  endstream endobj 591 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 39 /Filter /FlateDecode >> stream Hz##qoo+<<<,--wڥ ` 8 endstream endobj 592 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hz-${ endstream endobj 593 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H }@p2 endstream endobj 594 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H勮n__@ endstream endobj 595 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream Ȟ0gY endstream endobj 596 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H~z~~>@9 endstream endobj 597 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H255mhh0! endstream endobj 598 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hھ}֗/_ _ endstream endobj 599 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream Hz|`2P endstream endobj 600 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream HJLLLOO0Y endstream endobj 601 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream H:pOff&@G endstream endobj 602 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 45 /Filter /FlateDecode >> stream Haaa  endstream endobj 603 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 20 /Filter /FlateDecode >> stream HRQQa@ \ endstream endobj 604 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream H2eWzzz + endstream endobj 605 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H:vߵk o endstream endobj 606 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream HJLL08 endstream endobj 607 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream Huww𡱱?@:G endstream endobj 608 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H0D endstream endobj 609 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 28 /Filter /FlateDecode >> stream H󧟟ŋݻm6Q  endstream endobj 610 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream HJKK04M endstream endobj 611 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H;@ Ac endstream endobj 612 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 43 /Filter /FlateDecode >> stream Hf``~,X>| U endstream endobj 613 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hڹs2$p s | endstream endobj 614 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 23 /Filter /FlateDecode >> stream H󳴴 kjj0 endstream endobj 615 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream Ht钃Puu5@% endstream endobj 616 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HڲeK}}=@  endstream endobj 617 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream H ޻wpTT@6" endstream endobj 618 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H*//?|0@  endstream endobj 619 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream HhLLѣG ;d0 endstream endobj 620 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hz򥾾޽{  endstream endobj 621 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hjhh@ bo endstream endobj 622 0 obj << /Type /XObject /Subtype /Image /Width 10 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 42 /Filter /FlateDecode >> stream HϏTQQ铡444/^ ` j endstream endobj 623 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 25 /Filter /FlateDecode >> stream HzŋO:C, T endstream endobj 624 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 25 /Filter /FlateDecode >> stream HùsnܸqEC 6 endstream endobj 625 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HzYII @8 endstream endobj 626 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 39 /Filter /FlateDecode >> stream Hooo 9 endstream endobj 627 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 25 /Filter /FlateDecode >> stream HSΟ?OA T endstream endobj 628 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hz𡎎Ώ?  endstream endobj 629 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H*--;@ E endstream endobj 630 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 25 /Filter /FlateDecode >> stream Hzݻ;F endstream endobj 631 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream H l endstream endobj 632 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H k endstream endobj 633 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:~7@ q7 endstream endobj 634 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H7o^]]@ <U endstream endobj 635 0 obj << /Type /XObject /Subtype /Image /Width 1 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 15 /Filter /FlateDecode >> stream HZn@ endstream endobj 636 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HڵkWjj*@ J^ endstream endobj 637 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hڿ…  ! endstream endobj 638 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HJOO߲e @ GR endstream endobj 639 0 obj << /Type /XObject /Subtype /Image /Width 1 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 15 /Filter /FlateDecode >> stream Hz5@ endstream endobj 640 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H7o  endstream endobj 641 0 obj << /Type /XObject /Subtype /Image /Width 1 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 15 /Filter /FlateDecode >> stream H+@ endstream endobj 642 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 2 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream HyfMMUUU 3 endstream endobj 643 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 2 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream H~zQQѡCttt 2 endstream endobj 644 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 2 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream H:pʭ[ + endstream endobj 645 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:vشi $ endstream endobj 646 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 33 /Filter /FlateDecode >> stream H A7 endstream endobj 647 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HӧO  endstream endobj 648 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 15 /Filter /FlateDecode >> stream HZ   endstream endobj 649 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Ht˗/ 2 endstream endobj 650 0 obj << /Type /XObject /Subtype /Image /Width 13 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 54 /Filter /FlateDecode >> stream H'GGGFFF444DDD``` Dr endstream endobj 651 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hڶm[hh(@ " endstream endobj 652 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hntt4@~ endstream endobj 653 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HzUXX@ endstream endobj 654 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream H>w\ooUnܸ`[t 9 endstream endobj 655 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H:t萕U\\@ endstream endobj 656 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hx"@ o[ endstream endobj 657 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hڷoߓ'O  endstream endobj 658 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H0 3e endstream endobj 659 0 obj << /Type /XObject /Subtype /Image /Width 13 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 54 /Filter /FlateDecode >> stream H'䫫(((ӹڤ x endstream endobj 660 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H 0 oz endstream endobj 661 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H244ܴi@ endstream endobj 662 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hjoo?s @  endstream endobj 663 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 25 /Filter /FlateDecode >> stream HӧOݻsG endstream endobj 664 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H 0_ endstream endobj 665 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hqׯ 0J endstream endobj 666 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hwޓ'O?`' endstream endobj 667 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 25 /Filter /FlateDecode >> stream HӧOݻ F endstream endobj 668 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hڲe$pUp,  endstream endobj 669 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 26 /Filter /FlateDecode >> stream H铃++"ܿ E endstream endobj 670 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H6m @ 8w endstream endobj 671 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HZ`ɓ  endstream endobj 672 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H*++ 0e endstream endobj 673 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HZtiww7@  endstream endobj 674 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H8qŋ k' endstream endobj 675 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream Hz捡! o C50 endstream endobj 676 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H 0\ endstream endobj 677 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 39 /Filter /FlateDecode >> stream H촳Ӯ]444xyy_z` n endstream endobj 678 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream Hz򥎎 9@/] endstream endobj 679 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H1cF 8v@Z6 endstream endobj 680 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream H:q℡Ԟ={`I; endstream endobj 681 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H> stream Hrpphii0M endstream endobj 683 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H> stream HZj&@ z endstream endobj 685 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream Hھ}Ծ~ `F; endstream endobj 686 0 obj << /Type /XObject /Subtype /Image /Width 11 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 45 /Filter /FlateDecode >> stream Hx񢖖˗/kkk򂃃322ddd S endstream endobj 687 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream HڸqҥK 0b endstream endobj 688 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HæMZ[[ $v endstream endobj 689 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hq@Q endstream endobj 690 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H2330 `A endstream endobj 691 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream Hrwwomm0Me endstream endobj 692 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 32 /Filter /FlateDecode >> stream HzEppp``#jq endstream endobj 693 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HZb@  endstream endobj 694 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 33 /Filter /FlateDecode >> stream H { L endstream endobj 695 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream HHOO5k#@#/ endstream endobj 696 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream H:uꔦf@@@3M endstream endobj 697 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HzՋ/ ,w endstream endobj 698 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 28 /Filter /FlateDecode >> stream Hz̙ 999X\ endstream endobj 699 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 20 /Filter /FlateDecode >> stream H233c@.P endstream endobj 700 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream H> stream HRQQ),,0 endstream endobj 702 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream Hmnn0A endstream endobj 703 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H0  endstream endobj 704 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hڴi@ X endstream endobj 705 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 29 /Filter /FlateDecode >> stream H{;wjjjld ; endstream endobj 706 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream Hrss[nٳgM`:JW endstream endobj 707 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream Hjhhػw9@& endstream endobj 708 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream H7ow^^^@@; 6 endstream endobj 709 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream H655ݻw/@  endstream endobj 710 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream HrppرcDZc % endstream endobj 711 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 20 /Filter /FlateDecode >> stream HA0PU endstream endobj 712 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 39 /Filter /FlateDecode >> stream H/++6mCBBÇ p  endstream endobj 713 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H0a@ . endstream endobj 714 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream Hڻwu "/ endstream endobj 715 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H0 endstream endobj 716 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H2eʵk ? endstream endobj 717 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream HXrellܤI *N endstream endobj 718 0 obj << /Type /XObject /Subtype /Image /Width 14 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 55 /Filter /FlateDecode >> stream H񣲲rTTޕ+W+**ZZZvq̙kkk~ endstream endobj 719 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream HRTTܰaChh:@ endstream endobj 720 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 42 /Filter /FlateDecode >> stream Hڨ̭ "k endstream endobj 721 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:s̴i ~' endstream endobj 722 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 25 /Filter /FlateDecode >> stream H:s̒%K-[~7 endstream endobj 723 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HRSS۳g@ endstream endobj 724 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H>w@  endstream endobj 725 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream Hҧ _y endstream endobj 726 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 33 /Filter /FlateDecode >> stream H | ' endstream endobj 727 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 33 /Filter /FlateDecode >> stream H*//1cF__7o7l`v  endstream endobj 728 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 25 /Filter /FlateDecode >> stream Hڻwϟ?l2> L endstream endobj 729 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hv횮{  endstream endobj 730 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HzY\\ׯ_  endstream endobj 731 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HZdɍ7 u endstream endobj 732 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream HpBss3@H endstream endobj 733 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream Hzⅲrfffxx[ > endstream endobj 734 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H0  endstream endobj 735 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H6mZZZ@ 4 endstream endobj 736 0 obj << /Type /XObject /Subtype /Image /Width 7 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 32 /Filter /FlateDecode >> stream Hrpp핓hii0P endstream endobj 737 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H zh endstream endobj 738 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hs۷o x\ endstream endobj 739 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hĉϟ`# endstream endobj 740 0 obj << /Type /XObject /Subtype /Image /Width 3 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream HZx` endstream endobj 741 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 2 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream H={v___OO///@)G endstream endobj 742 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hڳg $B endstream endobj 743 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:qℊ @  endstream endobj 744 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 28 /Filter /FlateDecode >> stream H:pI:;;N7M 3 endstream endobj 745 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hڼyŋ U endstream endobj 746 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream HrʤI_x`Pj 6 endstream endobj 747 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H:p2@  endstream endobj 748 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream H0 a= endstream endobj 749 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream HZhQ__߄ ߿`/ endstream endobj 750 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream H;w n `.E endstream endobj 751 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 20 /Filter /FlateDecode >> stream HB۷o0EW endstream endobj 752 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream H G :}F endstream endobj 753 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hڻwڵk '? endstream endobj 754 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream H勁={ AR endstream endobj 755 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 23 /Filter /FlateDecode >> stream HZd b= endstream endobj 756 0 obj << /Type /XObject /Subtype /Image /Width 9 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 39 /Filter /FlateDecode >> stream Hrsssww:u$$$6m`2 - endstream endobj 757 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hz$ endstream endobj 758 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream HpBAA^z`` 6 endstream endobj 759 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 29 /Filter /FlateDecode >> stream H:Ç끌ϟw endstream endobj 760 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 30 /Filter /FlateDecode >> stream Hzy]]K= `eg 9 endstream endobj 761 0 obj << /Type /XObject /Subtype /Image /Width 12 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 51 /Filter /FlateDecode >> stream H$,,,111bbbZZZ??? f, endstream endobj 762 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream HZlYHHHqqŋ < endstream endobj 763 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 21 /Filter /FlateDecode >> stream Hڳg*$p-vV endstream endobj 764 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream H7o  endstream endobj 765 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hټy3@> endstream endobj 766 0 obj << /Type /XObject /Subtype /Image /Width 6 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 28 /Filter /FlateDecode >> stream Hqrr2 `?K endstream endobj 767 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 18 /Filter /FlateDecode >> stream Hڿyy9@ endstream endobj 768 0 obj << /Type /XObject /Subtype /Image /Width 2 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 17 /Filter /FlateDecode >> stream HRPPhoo0 endstream endobj 769 0 obj << /Type /XObject /Subtype /Image /Width 4 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 24 /Filter /FlateDecode >> stream H:x𠾾K$%% *9 endstream endobj 770 0 obj << /Type /XObject /Subtype /Image /Width 8 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 37 /Filter /FlateDecode >> stream HJLL011q'Oj*q endstream endobj 771 0 obj << /Type /XObject /Subtype /Image /Width 5 /Height 1 /BitsPerComponent 8 /ColorSpace 891 0 R /Length 27 /Filter /FlateDecode >> stream 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G?T}G_,V2%:9kE_8꫔eKFy/FnיVIU}/$0 o/7_ɲl9.C&YPm>{m<(  , Equation Equation.30,Microsoft Equation 3.0Equation Equation.30,Microsoft Equation 3.0 Equation Equation.30,Microsoft Equation 3.0 Equation Equation.30,Microsoft Equation 3.0 Equation Equation.30,Microsoft Equation 3.0`/ 0DArial** )(0)z[ 0)"DSymbol** )(0)z[ 0)@ .  @n?" dd@  @@`` | {D4 8 9   448844/X$2$Ƌ&Rϻ wH 0AA0 3f3@33wʚ;ʚ;g4YdYd )z[ 0hppp@ <4ddddL) 0*  <4BdBdL)< 0*t80___PPT10 ?  %M4(NP-Complete and Approximation AlgorithmsPolynomial Algorithms LProblems encountered so far are polynomial time algorithms The worst-case running time of most of these algorithms is O(nk) time, for some constant k. All problems cannot be solved in polynomial time There are problems that cannot be solved at all  Unsolvable There are problems that can be solved but not in O(nk) time for some constant time. Problems that are solvable in polynomial time by polynomial-time algorithms are said to be tractable (or easy or efficient). Problems that require superpolynomial time are said to be intractable or hard.'Pw  31 3=3bw      ) P and NPClass P problems are solvable in polynomial time. Class NP problems are verifiable in polynomial time. For example, given a problem, we can verify the solution in polynomial time Any problem in P is also in NP P NP It is NOT KNOWN whether P is a proper subset of NP.pgZLZYZgL!  3%  NP What are NP-complete Problems? A problem is said to be NP-Complete if it is as  hard as any problem in NP. No polynomial time algorithm has yet been discovered for an NP-Complete problem. However, it has not been proven that NO polynomial time algorithm can exist for an NP-Complete problem. This problem was first posed by Cook in 1971. The issue of, P=NP or P NP is an open research problemXP33M   Examples of NP-Complete problems!!    Shortest vs. longest simple paths Finding the shortest paths from a single source in a directed graph G =(V,E) can be completed in O(V E) time. Even with negative edge weights. However, finding the longest simple path between two vertices is NP-complete. It is NP-Complete even if each of edge weights is equal to one. An Euler tour of a connected directed graph G=(V,E), can be completed in O(E) time. However, the Hamiltonian Cycle is NP-Complete cycle. The traveling salesman problem is a variation of the Hamiltonian cycle. #PPPPTPP3T53H Polynomial Time Reductions     Decision Problems: Problems whose answer is yes or no!! Most problems can be converted to decision problems. Language recognition problem is a decision problem. Suppose L U is the set of all inputs for which the answer to the problem is yes  L is called the language corresponding to the problem (Turing machines). The terms language and problem are used interchangeably. Given a problem, with an input language X, Now the decision problem can be defined as the problem to recognize whether or not X belongs to L.T=P:PP8sY  Reductions Contd.  &Definition: Let L1 and L2 be two languages from input spaces U1 and U2. We say that L1 is polynomially reducible to L2 if there exists a polynomial-time algorithm that converts each input u1 U1 to another input u2U2 such that u1 L1 iff u2 L2. The algorithm is polynomial in the size of the input u1. If we have an algorithm for L2 then we can compose the two algorithms to produce an algorithm for L1. If L1 is polynomially reducible to L2 and there is a polynomial-time algorithm for L2, then there is a polynomial time algorithm for L1. Reducibility is not symmetric L1 is polynomially reducible to L2 does not imply L2 is polynomially reducible to L1P333333 3333  Z           &    !!The Satisfiability (SAT) Problem  "   .   &S  Boolean expression in Conjunctive Normal Form (CNF) (Product (AND) of Sums (ORs) For example S = (x1+x2+x3)( x1+x2+x3)(x1+x2+x3) The SAT problem is to determine whether a given Boolean expression is Satisfiable (without necessarily finding a satisfying assignment) We can guess a truth assignment and check that it satisfies the expression in polynomial time. SAT is NP hard A Turing machine and all of its operations on a given input can be described by a Boolean Expression. The expression will be satisfiable iff the Turing machine will terminate at an accepting state for the given input. http://www.nada.kth.se/~viggo/problemlist/compendium.html8PP>PV L 3Q  {   "     V 2 Clique Problem Clique Problem  Clique ProblemClique Problem Clique ProblemVertex Cover problem Vertex Cover problem  Vertex Cover problem Vertex Cover problem Vertex Cover problem Dominating SetG = (V,E) is an undirected graph. A dominating set D of G is a set of vertices (a subset of V) such that every vertex of G is either in D or is adjacent to at least one vertex from D.b3S, Dominating SetG = (V,E) is an undirected graph. A dominating set D of G is a set of vertices (a subset of V) such that every vertex of G is either in D or is adjacent to at least one vertex from D.b3S, Dominating SetG = (V,E) is an undirected graph. A dominating set D of G is a set of vertices (a subset of V) such that every vertex of G is either in D or is adjacent to at least one vertex from D.b3S,Dominating SetG = (V,E) is an undirected graph. A dominating set D of G is a set of vertices (a subset of V) such that every vertex of G is either in D or is adjacent to at least one vertex from D.b3S,Dominating SetG = (V,E) is an undirected graph. A dominating set D of G is a set of vertices (a subset of V) such that every vertex of G is either in D or is adjacent to at least one vertex from D.b3S,!Dealing with NP Complete problems"" vProving that a given problem is NP-Complete does not make the probem go away!! Udi Manber An NP-Complete problem cannot be solved precisely in polynomial time We make compromises in terms of optimality, robustness, efficiency, or completeness of the solution. Approximation algorithms do not lead to optimal solutions Probabilistic algorithms Branch and bound Backtracking nf:8OPf:8>>  Backtracking  2The Hamiltonian Circuit Problem The Subset problemBranch and BoundJob Assignment Knapsack   ` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> f(    6LƝ   `}   T Click to edit Master title style! !  0ȝ   `   RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0ϝ  ^ `   >*  0  ^    @*  04  ^ `   @*H  0޽h ? 3380___PPT10.S@' Default Design}  $(  r  S Hkp > p  r  S lp  `   p  H  0޽h ? 33___PPT10i.S'+D=' *= @B +$  pd$(  dr d S r  `}   r d S u  `   H d 0޽h ? 3380___PPT10.]$  h$(  hr h S p  `}   r h S L  `   H h 0޽h ? 3380___PPT10.@b$  l$(  lr l S   `}   r l S }  `   H l 0޽h ? 3380___PPT10.0m$  p$(  pr p S \W  `}   r p S h9  `   H p 0޽h ? 3380___PPT10.0$  t$(  tr t S   `}   r t S g.  ` .  H t 0޽h ? 3380___PPT10.#$  x$(  xr x S <  `}   r x S ,>.  ` .  H x 0޽h ? 3380___PPT10.@~}L$  |$(  |r | S <  `}   r | S  .  ` .  H | 0޽h ? 3380___PPT10.Lgn  }04C 0 (   x  c $p  `}  p    0 NA ? ?`  p  8 Pg  Pg`2   0PP`2   0 P `2   0 P@   04p  3x 2   0[ P 7  3y 2G@  g   g   0p  g 3z 2TB   c $D0  ^2  6  ^2  6  ^2  6 p  6Dp   g  3y 2  6tp    3z 2MF  g '    (  00p  g 3x 2TB )  c $D0  LB * c $D 8   7  UN  g       0(p  g 3z 2TB   c $D0  f2   6 Pf2   6   f2   6  @UN  g +   ,  0p  g 3x 2TB -  c $D0   .  0pp P 7  1y 2LB / c $D P  LB 0 c $D P  LB 1 c $D P LB 2 c $D P LB 3 c $D3P LB 4 c $D3`P  LB 5 c $D3 P P LB 6 c $D3 P @LB 8 c $D3 P LB 9 c $DP P LB : c $D P LB ; c $DP LB < c $D`   LB = c $D P  LB > c $DP  LB ? c $DP  LB @ c $Df3`  LB A c $Df3  LB B c $D` LB C c $D H  0޽h ? 33___PPT10i.S`a+D=' *= @B +r  47$(  $~ $ s *,p  `}  p   $0 TA ? ?`  p )F Pg $ Pg`2 $ 0PP`2 $ 0 P `2 $ 0 P@ $ 0Pp  3x 2  $ 0p P 7  3y 2UN  g  $  g  $ 0p  g 3z 2TB  $ c $D0  X2  $ 0  X2 $ 0  X2 $ 0 p $ 0p   g  3y 2 $ 0p    3z 2MF  g $   $ 0p  g 3x 2TB $ c $D0  LB $ c $D F   $  UN  g $   $ 0p  g 3z 2TB $ c $D0  `2 $ 0 P`2 $ 0   `2 $ 0  @UN  g $  $ 0Dw  g 3x 2TB $ c $D0    $ 0w P 7  1y 2LB !$ c $D P  LB "$ c $D P  LB #$ c $D P LB $$ c $D P LB %$ c $D3P LB &$ c $D3`P  LB '$ c $D3 P P LB ($ c $D3 P @LB )$ c $D3 P RB *$ s *D3ԔP P RB +$ s *D3Ԕ P LB ,$ c $DP LB -$ c $D`   LB .$ c $D P  LB /$ c $DP  LB 0$ c $DP  LB 1$ c $Df3`  RB 2$ s *D3Ԕ  LB 3$ c $D` LB 4$ c $D H $ 0޽h ? 33___PPT10i.S`a+D=' *= @B +  77<9(  <~ < s *8w  `}  w   <0 TA ? ?`  w )F Pg < Pg`2 < 0PP`2 < 0 P `2 < 0 P@ < 0dw  3x 2  < 0w P 7  3y 2UN  g  <  g  < 0w  g 3z 2TB  < c $D0  X2  < 0  X2 < 0  X2 < 0 p < 0 w   g  3y 2 < 0$w    3z 2MF  g <   < 0(w  g 3x 2TB < c $D0  LB < c $D F   <  UN  g <   < 0-w  g 3z 2TB < c $D0  `2 < 0 P`2 < 0   `2 < 0  @UN  g <  < 0x2w  g 3x 2TB < c $D0    < 06w P 7  1y 2LB !< c $D P  LB "< c $D P  LB #< c $D P LB $< c $D P LB %< c $D3P LB &< c $D3`P  LB '< c $D3 P P LB (< c $D3 P @LB )< c $D3 P RB *< s *D3ԔP P RB +< s *D3Ԕ P LB ,< c $DP LB -< c $D`   LB .< c $D P  LB /< c $DP  LB 0< c $DP  LB 1< c $Df3`  RB 2< s *D3Ԕ  LB 3< c $D` LB 4< c $D XB 5< 0D3oXB 6< 0D3o@ @ 0XB 7< 0D3oH < 0޽h ? 33___PPT10i.S`a+D=' *= @B +  78(9(  (~ ( s *xEw  `}  w   (0 TA ? ?`  w )F Pg ( Pg`2 ( 0PP`2 ( 0 P `2 ( 0 P@ ( 0Hw  3x 2  ( 0Lw P 7  3y 2UN  g  (  g  ( 0Jw  g 3z 2TB  ( c $D0  X2  ( 0  X2 ( 0  X2 ( 0 p ( 0Tw   g  3y 2 ( 0Yw    3z 2MF  g (   ( 0\w  g 3x 2TB ( c $D0  LB ( c $D F   (  UN  g (   ( 0law  g 3z 2TB ( c $D0  `2 ( 0 P`2 ( 0   `2 ( 0  @UN  g (  ( 0fw  g 3x 2TB ( c $D0    ( 0jw P 7  1y 2LB !( c $D P  LB "( c $D P  LB #( c $D P LB $( c $D P LB %( c $D3P RB &( s *D3Ԕ`P  LB '( c $D3 P P RB (( s *D3Ԕ P @LB )( c $D3 P LB *( c $DP P LB +( c $D P LB ,( c $DP LB -( c $D`   LB .( c $D P  LB /( c $DP  LB 0( c $DP  LB 1( c $Df3`  LB 2( c $Df3  RB 3( s *DԔ` LB 4( c $D XB 5( 0D3Ԕ000XB 7( 0D3Ԕ `XB 8( 0D3Ԕ0H ( 0޽h ? 33___PPT10i.S`a+D=' *= @B +  77@9(  @~ @ s *lw  `}  w   @0 TA ? ?`  w )F Pg @ Pg`2 @ 0PP`2 @ 0 P `2 @ 0 P@ @ 0ؑw  3x 2  @ 0w P 7  3y 2UN  g  @  g  @ 0w  g 3z 2TB  @ c $D0  X2  @ 0  X2 @ 0  X2 @ 0 p @ 0w   g  3y 2 @ 0xw    3z 2MF  g @   @ 0w  g 3x 2TB @ c $D0  LB @ c $D F   @  UN  g @   @ 0w  g 3z 2TB @ c $D0  `2 @ 0 P`2 @ 0   `2 @ 0  @UN  g @  @ 0w  g 3x 2TB @ c $D0    @ 0 w P 7  1y 2LB !@ c $D P  LB "@ c $D P  LB #@ c $D P LB $@ c $D P LB %@ c $D3P RB &@ s *D3Ԕ`P  LB '@ c $D3 P P RB (@ s *D3Ԕ P @LB )@ c $D3 P LB *@ c $DP P LB +@ c $D P LB ,@ c $DP LB -@ c $D`   LB .@ c $D P  LB /@ c $DP  LB 0@ c $DP  LB 1@ c $Df3`  LB 2@ c $Df3  RB 3@ s *D3Ԕ` LB 4@ c $D XB 5@ 0D3Ԕ000XB 6@ 0D3Ԕ `XB 7@ 0D3Ԕ`H @ 0޽h ? 33___PPT10i.S`a+D=' *= @B +~  P%(  r  S   `@   2  6 p 1A2  6t    1E2  6  `P  1D2  6, P@ 1C2   6 p` 1BLB  c $D@0LB  c $D@LB  c $D`PLB  c $D LB @ c $D LB  c $DP PP LB @ c $Dp@    <x" V A vertex cover of G = (V,E) is a set of vertices such that every edge in E is incident to at least one of the vertices in the vertex cover.H  0޽h ? 33___PPT10i.U@LQ+D=' *= @B +   #  , (  ,x , c $*  `@   2 , 6\, p 1A2 , 6/    1E2 , 63  `P  1D2 , 6p6 P@ 1C2 , 69 p` 1B2 , 6(= 0  1A2  , 6@    1E2  , 6C    1D2  , 6138 (  8x 8 c $Pf  `   x 8 c $$g  `@   RB 8 s *DP@PRB 8 s *DP@@@ RB 8 s *D@ @@ RB  8 s *DP@ X2  8 03 pX2  8 03  p X2  8 0 pp X2  8 0  RB 8 s *Dp  8 6k 0p 1A 2 8 6p    1D 2 8 6s 0  1B 2 8 6\w p W  1C 2 8 6z 0p 1A 2%F  0  8 0p 0 .N  0  8  0 TB 8 c $D@PPTB 8 c $D@P@@ TB 8 c $D@@ @ TB 8 c $DP@ Z2 8 s * pZ2 8 s * p Z2 8 s * pp Z2 8 s *  TB 8 c $Dp  8 0| 0p 1A 2  8 0     1D 2 !8 0,  0 1B 2 "8 0  p W  1C 2 #8 0, 0p 1A 2RB $8 s *D1p PRB %8 s *DjJp@PLB &8@ c $DP ` LB '8 c $D` @ LB (8 c $D@ P LB )8 c $Dp @P LB *8 c $DP@ 0 LB +8 c $D0 p @ LB ,8 c $DP LB -8 c $D @@  .8 0H @g 2AB 2 /8 0䙾  P  2AC 2 08 0H  @ 2CD 2 18 0    2BD 2 28 0 P p7  2AD 2 38 0|    NG has a vertex cover of size m 2H 8 0޽h ? 33___PPT10i.V0u+D=' *= @B +o  ~033L(  Lx L c $  `   x L c $  `@   %F  0  L 0  .N  0  L  0 TB L c $D@PPTB L c $D@P@@ TB L c $D@@ @ TB  L c $DP@ Z2  L s *3 pZ2  L s *3 p Z2  L s * pp Z2  L s *  TB L c $Dp  L 0L 0p 1A 2 L 0     1D 2 L 0þ  0 1B 2 L 0|Ǿ  p W  1C 2 L 0ʾ 0p 1A 2%F  0  L 0p 0 .N  0  L  0 TB L c $D@PPTB L c $D@P@@ TB L c $D@@ @ TB L c $DP@ Z2 L s *3 pZ2 L s *3 p Z2 L s * pp Z2 L s *  TB L c $Dp  L 0xо 0p 1A 2  L 0$Ծ     1D 2 !L 0p׾  0 1B 2 "L 0ھ  p W  1C 2 #L 0@޾ 0p 1A 2RB $L s *D1p PRB %L s *DjJp@PLB &L@ c $DP ` LB 'L c $D` @ LB (L c $D@ P LB )L c $Dp @P LB *L c $DP@ 0 LB +L c $D0 p @ LB ,L c $DP LB -L c $D @@  .L 0 @g 2AB 2 /L 0T  P  2AC 2 0L 0  @ 2CD 2 1L 0    2BD 2 2L 0 P p7  2AD 2. 3L 0   a G has a dominating set of size m if and only if G has a vertex cover of size mP 2PH L 0޽h ? 33___PPT10i.V0u+D=' *= @B +o  ~@33P(  Px P c $,  `   x P c $  `@   %F  0  P 0  .N  0  P  0 TB P c $D@PPTB P c $D@P@@ TB P c $D@@ @ TB  P c $DP@ Z2  P s * pZ2  P s * p Z2  P s * pp Z2  P s *  TB P c $Dp  P 0 0p 1A 2 P 0H     1D 2 P 0\  0 1B 2 P 0  p W  1C 2 P 0\ 0p 1A 2%F  0  P 0p 0 .N  0  P  0 TB P c $D@PPTB P c $D@P@@ TB P c $D@@ @ TB P c $DP@ Z2 P s * pZ2 P s * p Z2 P s * pp Z2 P s *  TB P c $Dp  P 0h 0p 1A 2  P 0$     1D 2 !P 0`'  0 1B 2 "P 0*  p W  1C 2 #P 00. 0p 1A 2RB $P s *D1p PRB %P s *DjJp@PLB &P@ c $DP ` LB 'P c $D` @ LB (P c $D@ P LB )P c $Dp @P LB *P c $DP@ 0 LB +P c $D0 p @ LB ,P c $DP LB -P c $D @@  .P 03 @g 2AB 2 /P 0D8  P  2AC 2 0P 0;  @ 2CD 2 1P 0?    2BD 2 2P 0xB P p7  2AD 2. 3P 0E   a G has a dominating set of size m if and only if G has a vertex cover of size mP 2PH P 0޽h ? 33___PPT10i.V0u+D=' *= @B +}  p$(  r  S U  `p   r  S V  `   H  0޽h ? 33___PPT10i.Y4|J+D=' *= @B +  PT0(  Tx T c $c  `}   x T c $d  `   H T 0޽h ? 33___PPT10i.ЙHo+D=' *= @B +}  `X$(  Xr X S bp  `}  p  r X S cp  ` p  H X 0޽h ? 33___PPT10i.д+D=' *= @B +zxMkAͬvIR=6%Gۤ&) A+O,Hԛ ofv)H5y/[V>j %p("#T2^!+E&R5*X je0I|Ҡ$p\i)+l*`]ɜC;P͆j?}':D^[mC{22'!J6AJ*+b` f ;{H{EGQ>[M($qdzZȾ?n0h7ۃ oqWk50?Ρ 3Ϻy l3VY}XhB݈ j5gO\xT3AyKXtp-˔F*)х!%8K Yi] Fb);Fn*2h0NMb;9`Ľ \#%e*eU)HٲGb~)sҹrp_cx$#}$'`R&3SCx<ljwN݌2Uͮo\v~HEQ>~ęfM?Z3.&?v_v{e/Pww4O s!MW}Qs_>Dk73~q-&kTg_U<jd3yymxs4JGSa'\L}K3+:G'ʨƽ/禓DZ Ʉ[QS/dza)+;iEU Bs|T,=ߍNjf٧5k33YSX 6򗘄jDFvxMOA,cYhia`@"!ŋ(&&@1%bh XJ4AH<ƻFœFUcPJ7<ҭ`m^f{>vw]G l"Wc2lXbE3D !WL(S/Xrs&R-jocsg? =|ɨy$ pn۠g1<&ЏP}?ڧz5QkέC:`|ɹo^ +Aj _IfA@E]t&Fp~n[E/پz<@e* Ph "zA|@P XM_~qŊSaLki| f_m1%F ;/Vd5j^%Y%n:YLFvFF2d $U\d_v`m= Er/4Z=2!w0 j1'8?r'r;+{/{izhɫx^(^meLfo8i"(,ZJrCÄ435DW8=N~|XAq1~s&@;vN/36m8NN}p H ;7!yÆ~o.sui6NI߯!zqjKT_Ӝ4nE."\ȳDox&rq<'tׄ2 ~x|)̬W?pMDÎV}{`CعX<60&CXlɪNX;xҭ_bԮuxMOA,cYRa`@"!ŋ(&&P rk(Z,% $]I1(%7<`Lh}ofgV0[CBj6/3c}Ý\ T"Wc\Xb %D  WL !]X ; fM֥[(VC)O"#uƇO?x{$ Hn۠gŞ1< &яQ8_ڧz5 c֜[;t=4Y(>^suhUHW2S0a(ցE]p&Fp~n[E/Jپz<@e Ph !z} >Ak,׌ҁ/=léo0>(c3YW:pLQszYf%W jΡfgm񆛺}jeV,Q #Y{ͨ)Ez W3--Ĭ5YZ[#C"~wK}ۘ^.3t ' ' A~5G ]S7(7Hgu&lqEćx,<K/SD3|0>= -mO+6^3*)R^+?./y~6sDmGMN=SgS8ܩ RcqHBƈsq\ڢöw^Z)\/V:4<$[ѫl~`H(vULBS<[N>3D)08n}?EN35NFfwgheGmHVEtvځGSn+E uxMOA,cYhia h@qk(ZlK4AH<ƻFœFQHLh}ofgV0[CBjx1쾏]z_|a'(A6Wy<Xl.#ۂ_vb 6 !WL1'!]X ۰ fM֥[(FC)O"#Gu}>%2<aźm{z (QG?|hz}nG9߇w> q{h_V}&z +[BL0#cXz-ꂫ41s7*P;=9*TUWXjD j\ƭQ:$x8FԋƘ϶.Ji1jܙnV϶"W*!WM;,qاVVf2Up 0_RWq5]\|rBYū >ė;˽h\ʄ #42cHǀP`ziȩ\=L!Yg=gAW&-zo4xA:>V3ao0 pECc0j!A43I7~3 ִb+S>Tᔿ8@ cE5Tv휨#:m-^S6m8wNN}rH5;7F!yC~w.sui6NI߫!z~jKX_ќ nE."\ȳDox&rq<'QtW2 ~x|!Y̬W?PuDÎV|{`COd,I]SD$6fvwVvddUD{'4/Rߒ~^|xMOA,c)EpX=Hxb4 TqkXJ4AH<ƻœFUcPJ?&fޛӅu7"UP^c\^ybiMP1NlºA/$:!(dK@~N9R5ldk7WCdHmO{ƨL$1BT-ЁR_e? +پST?9{ɇ! `|}ɹzn^ +A& #0!A@Ф;J#p?q{ék پ<@e* PFAE@P Y364%3N|z1tfM܊cFw曞S-ab&ljx$23VwTm+M$k$H^2~MBJ^Lvqq帅Wm_ P,@s)88iPeƐCa,="rLw3wf~=t)[W75yoޣxQdNZƄ-H,4ћ0|.E4i/~ѳҖlKn|.JS~a²sE7{.]n^a7\v/_sy2D1osq\]>ڢղwj].Xi/4<$ѫl^`$SQ:l+B>xd.]O_ x0xRÎf=0;rf4L ΑP*mx"YKzOV4rz<xF1X<  ~n!Uu/"*BwZrsuxH)KUMOQSV? ! Oh+'0 `h  )NP-Complete and Approximation Algorithmsft  Mohan Kumar Mohan Kumar5haMicrosoft PowerPointxim@EmB@S@pUdAGjg  +  -- @ !--'@Arial-.  2 +NP,)."System-@Arial-.  2 +V-.-@Arial-. 2 +j Complete and ,"3"""""".-@Arial-. +2 uApproximation Algorithms)"""3""")"""3.-՜.+,0H    On-screen ShowCSE@UTAA ArialSymbolDefault DesignMicrosoft Equation 3.0)NP-Complete and Approximation AlgorithmsPolynomial Algorithms P and NPNP !Examples of NP-Complete problemsPolynomial Time Reductions Reductions Contd. "The Satisfiability (SAT) Problem Clique ProblemClique ProblemClique ProblemClique ProblemClique ProblemVertex Cover problemVertex Cover problemVertex Cover problemVertex Cover problemVertex Cover problemDominating SetDominating SetDominating SetDominating SetDominating Set"Dealing with NP Complete problems BacktrackingBranch and Bound  Fonts UsedDesign TemplateEmbedded OLE Servers Slide Titles#_㗌  Mohan KumarMohan Kumar  !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root EntrydO)PicturesCurrent UserSummaryInformation(PowerPoint Document(DocumentSummaryInformation8Root EntrydO) QPicturesCurrent UserJSummaryInformation(&_㗌Savita ChauhanSavita Chauhancse5311-old/Notes.htm010064400020320000200000000067561013365252600147650ustar00cse5311cse531100000400000024 Notes and Exercises
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Module1-Class Notes      Module1-Additional Notes Exercise1
Module2-Class Notes      Module2-Additional Notes Exercise2
Module3-Class Notes      Module3-Additional Notes Exercise3
Module4-Class Notes      Exercise4
Module5-Class Notes      Exercise5
Module6-Class Notes      Exercise6
Module7-Class Notes      Exercise7
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cse5311-old/References.htm010064400020320000200000000201361003507357100157400ustar00cse5311cse531100000400000024 References
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References:   (List is incomplete)

 

§         The Design and Analysis of Algorithms 1974

AV Aho, JE Hopcroft and JD Ullman, Addison-Wesley Publishing Company

 

§         Introduction to Algorithms: A Creative Approach, Reprinted 1989

Udi Manber,   Addison-Wesley Publishing Company

 

§         Introduction to Algorithms, 1982

Sedgewick, Addison Wesley Publishing Company

 

§         Graph Algorithms, 1979

Shimon Even, Computer Science Press

 

§         Introduction to the Theory of Computation, 1992

Michael Sipser, PWS Publishing Company

 

§         The Art of Computer Programming, Vols. 1 and 3

Knuth, Addison Wesley Publishing Company

Arial">References
:   (List is incomplete)

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[1]  R.L. Wainwright, A Class of Sorting Algorithms based on Quicksort, Communications of the ACM, Vol. 28, No. 4, April 1985, pgs. 396-402.

[2] R.S. Scowen, Algorithm 271: Quicksort, Communications of the ACM, Vol. 8, No. 11, Nov. 1965, pgs. 669-670.

[3] M.N. vanEmden, Algorithm 402: Increasing the efficiency of Quicksort, Communications of the ACM, Vol. 13, No. 11, Nov. 1970, pgs. 693-694.

[4] C.R. Cook, and Kim D.J, Best sorting algorithm for nearly sorted lists, Communications of the ACM, Vol. 23, No. 11, Nov. 1980, pgs. 620-624.

[5] C.A.R. Hoare, Algorithm 64:Quicksort, Communications of the ACM, Vol. 4, No. 7, July 1961, pg. 321.

 
cse5311-old/Statement of Ethics.doc010064400020320000200000000740001003507357200173650ustar00cse5311cse531100000400000024ࡱ> 7965@ Pbjbj22 3 XXP   m  $_Rl  '   `̤m "=0m.$, t r \B M  Statement of Ethics Student Confirmation (CSE 1325-001, Spring 2002) The following is an excerpt from the College of Engineerings statement on Ethics, Professionalism, and Con-duct of Engineering Students. The notes are modifications appropriate for Computer Science and Engineering courses. Read the statement carefully, sign it, and return it to your instructor. A copy of the original policy is available for examination in the Computer Science and Engineering office. Additional copies of this statement can be obtained from your instructor or the Computer Science and Engineering office. Statement on Ethics, Professionalism, and Conduct of Engineering Students College of Engineering The University of Texas at Arlington The College cannot and will not tolerate any form of academic dishonesty by its students. This includes, but is not limited to 1) cheating on examination, 2) plagiarism, or 3) collusion. Definitions: Cheating on an examination includes: Copying from anothers paper, any means of communication with another during an examination, giving aid to or receiving aid from another during an examination; Using any material during an examination that is unauthorized by the proctor; Taking or attempting to take an examination for another student or allowing another student to take or attempt to take an examination for oneself. Using, obtaining, or attempting to obtain by any means the whole or any part of an unadministered examination. Plagiarism is the unacknowledged incorporation of anothers work into work which the student offers for credit. Collusion is the unauthorized collaboration of another in preparing work that a student offers for credit. D. Other types of academic dishonesty include using other students printouts from the ACS labs or students disk, etc. Notes: The use of the source code of another persons program, even temporarily, is considered plagiarism. Allowing another person to use your source code, even temporarily, is considered collusion. In this class, the specific exceptions given below are not considered scholastically dishonest acts: Discussion of the algorithm and general programming techniques used to solve a problem Giving and receiving aid in debugging Discussion and comparison of program output The penalty assessed for cheating on a given assignment will be twice the weight of the assignment and will include notification of the proper authorities as stipulated in the UTA Handbook of Operating Procedures and on the web at  HYPERLINK "http://www2.uta.edu/discipline" http://www2.uta.edu/discipline You may be entitled to know what information UT Arlington (UTA) collects concerning you. You may review and have UTA correct this information according to procedures set forth in UT System BPM #32. The law is found in sections 552.021, 552.023 and 559.004 of the Texas Government Code. I have read and I understand the above statement. 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How would you extend the Rabin-Karp method to handle the problem of searching a text string for an occurrence of any of a given set of k patterns? The input is two strings of characters A = a1 a2 . . . an and B = b1 b2 . . . bn. Design an O(n) algorithm to determine whether B is a cyclic shift of A. In other words, the algorithm should determine whether there exists an index k, 1( k( n such that ai = b(k+i) mod n, for all i, 1( i( n. 4. The LCS of two sequences T and P is the largest sequence L such that L is a subsequence of both T and P. The smallest common supersequence (SCS) of two sequences T and P is the smallest sequence L such that both T and P are subsequences of L . a) Design an efficient algorithm to find LCS and SCS of two sequences b) Let d(t,p) be the smallest edit distance between T and P such that no replacements are allowed (in other words we have to insert and delete). Prove that d(T,P) = |SCS(T,P) - |LCS(T,P)| , where |SCS(T,P) | (|LCS(T,P) |) is the size of the smallest SCS (LCS) of T and P. 5. Let A = a1 a2 . . . an and B = b1 b2 . . . bm bet two strings of characters. Denote by A[i] the string ai ai+1. . . an, (namely the ith suffix of A). Let di be the minimal edit distance between B and A[i]. Design an O(n2) algorithm to find the minimum value of di (among all i). 6. The input is d sequences of elements such that each sequence is already sorted, and there is a total of n elements. 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KMP Algorithm4Knuth-Morris-Pratt Algorithm Utilize a prefix array  [1..m], where  [q] contains information to compute (q, a) for (a T " ), the pattern shift for a mismatch on P [q].  requires only O (m) time (as opposed to O (m |"|) for ). Pseudocode : Compute-Prefix-Function (P) m = length (P)  [1] = 0 ; k must be less than q k = 0 for q = 2 to m ; O (m) amortized while k > 0 and P[k+1] `" P [q] k =  [k] if P[k+1] = P [q] then k = k + 1 ; prefix increased by one  [q] = k return PP-P" " &""  $"$0((,",.004"4 88 88/88<"<v" " O"" 6&4yO KMPPseudocode : KMP-Matcher (T, P) n = length (T) m = length (P)  = Compute-Prefix-Function (P) ; O (m) amortized q = 0 for i = 1 to n ; O (n) amortized while q > 0 and P[q+1] `" T [i] ; where do we move to in P? q =  [q] if P[q+1] = T [i] ; matches so far then q = q + 1 if q = m then print ``Pattern occurs with shift'' (i - m) q =  [q] This algorithm takes O (m + n) timeP$P7" " -""*   GJE+  ` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> :2 H(  H H 6(1  " `}   T Click to edit Master title style! !$ H 0  " `   RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S H 0L  "^ `   H*0  H 0  "^    J* 0  H 0  "^ `   J* 0 H H 0޽h ? 3380___PPT10.+I>8 Default Design 0 zr00 ( \b 0 0 0Y  P   T  P*   0 0Y     T  R*  d 0 c $ ?  Y  0 0Y   @ Y  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S 0 6Y  `P  Y  P*   0 6Y  `  Y  R*  H 0 0޽h ? ̙3380___PPT10.*ty  @$(  r  S  H   r  S  H@`     H  0޽h ? ̙33`  P (   h  0T  In string matching problems, it is required to find the occurrences of a pattern in a text. These problems find applications in text processing, text-editing, computer security, and DNA sequence analysis. Find and Change in word processing Sequence of the human cyclophilin 40 gene CCCAGTCTGG AATACAGTGG CGCGATCTCG GTTCACTGCA ACCGCCGCCT CCCGGGTTCA AACGATTCTC CTGCCTCAGC CGCGATCTCG : DNA binding protein GATA-1 CCCGGG : DNA binding protein Sma 1 C: Cytosine, G : Guanine, A : Adenosine, T : Thymine& 2=  5H  0޽h ? ̙33  ~v`( 8 :   0 /T  vText : T[1..n] of length n and Pattern P[1..m] of length m. The elements of P and T are characters drawn from a finite alphabet set S. For example S= {0,1} or S = {a,b, . . . , z}, or S = {c, g, a, t}. The character arrays of P and T are also referred to as strings of characters. Pattern P is said to occur with shift s in text T if 0 s n-m and T[s+1..s+m] = P[1..m] or T[s+j] = P[j] for 1 j m, such a shift is called a valid shift. The string-matching problem is the problem of finding all valid shifts with which a given pattern P occurs in a given text T. H 2  J :,|H  0޽h ? ̙33  p"( LM r  S T H   T  p  0XET 0q To find all valid shifts or possible values of s so that P[1..m] = T[s+1..s+m] ; There are n-m+1 possible values of s. Procedure BF_String_Matcher(T,P) 1. n length [T]; 2. m length[P]; 3. for s 0 to n-m 4. do if P[1..m] = T[s+1..s+m] 5. then shift s is valid This algorithm takes Q((n-m+1)m) in the worst case. bx5 2x   H  0޽h ? ̙33  W(    0$pT  0 is a c a a b c a c a a b c a a b a a b a c a a b c a a b a c a a b c matches a a b l 2  " jB  BDjB  BD0 jB  BD 00 H  0޽h ? ̙33L  (  r  S T H   T    0T 0 zLet S = {0,1,2, . . .,9}. We can view a string of k consecutive characters as representing a length-k decimal number. Let p denote the decimal number for P[1..m] Let ts denote the decimal value of the length-m substring T[s+1..s+m] of T[1..n] for s = 0, 1, . . ., n-m. ts = p if and only if T[s+1..s+m] = P[1..m], and s is a valid shift. p = P[m] + 10(P[m-1] +10(P[m-2]+ . . . +10(P[2]+10(P[1])) We can compute p in O(m) time. Similarly we can compute t0 from T[1..m] in O(m) time.7 2 g  H  0޽h ? ̙33=  4}(  4 4 0T    s9p = P[m] + 10(P[m-1] +10(P[m-2]+ . . . +10(P[2]+10(P[1])):9 4 00T r l6378 = 8 + 7 10 + 3 102 + 6 103 = 8 + 10 (7 + 10 (3 + 10(6))) = 8 + 70 + 300 + 6000nr 2   ? 4 0ȩT p@ 6m =4 2H 4 0޽h ? 3   (     0T p xts+1 can be computed from ts in constant time. ts+1 = 10(ts  10m-1 T[s+1])+ T[s+m+1] Example : T = 314152 ts = 31415, s = 0, m= 5 and T[s+m+1] = 2 ts+1= 10(31415  10000*3) +2 = 14152 Thus p and t0, t1, . . ., tn-m can all be computed in O(n+m) time. And all occurences of the pattern P[1..m] in the text T[1..n] can be found in time O(n+m). However, p and ts may be too large to work with conveniently. Do we have a simple solution!!`     ) ) -     .<      A   `H  0޽h ? ̙33R    $(  $Z $ 0T  zComputation of p and t0 and the recurrence is done modulus q. In general, with a d-ary alphabet {0,1,& ,d-1}, q is chosen such that dq fits within a computer word. The recurrence equation can be rewritten as ts+1 = (d(ts  T[s+1]h)+ T[s+m+1]) mod q, where h = dm-1(mod q) is the value of the digit  1 in the high order position of an m-digit text window. Note that ts p mod q does not imply that ts = p. However, if ts is not equivalent to p mod q , then ts p, and the shift s is invalid. We use ts p mod q as a fast heuristic test to rule out the invalid shifts. Further testing is done to eliminate spurious hits. - an explicit test to check whether P[1..m] = T[s+1..s+m] B 2 oO  _    (  H $ 0޽h ? ̙33  NF8( l 8 8 0T p@ ( 2 8 0T   ts+1 = (d(ts  T[s+1]h)+ T[s+m+1]) mod q h = dm-1(mod q) Example : T = 31415; P = 26, n = 5, m = 2, q = 11 p = 26 mod 11 = 4 t0 = 31 mod 11 = 9 t1 = (10(9 - 3(10) mod 11 ) + 4) mod 11 = (10 (9- 8) + 4) mod 14 = 14 mod 11 = 3 2 H 8 0޽h ? 3     ( ( [ ( ( 0 Y  Procedure RABIN-KARP-MATCHER(T,P,d,q) Input : Text T, pattern P, radix d ( which is typically =S), and the prime q. Output : valid shifts s where P matches 1. n length[T]; 2. m length[P]; 3. h dm-1 mod q; 4. p 0; 5. t0 0; 6. for i 1 to m 7. do p (dp + P[i] mod q; 8. t0 (dt0 +T[i] mod q; 9. for s 0 to n-m 10. do if p = ts 11. then if P[1..m] = T[s+1..s+m] 12. then  pattern occurs with shift  s 13. if s < n-m 14. then ts+1 (d(ts  T[s+1]h)+ T[s+m+1]) mod q; nJ 2 !4            H ( 0޽h ? ̙33  `X ,(  ,r , S Y H   T  > , <UY J  $All characters are interpreted as radix-d digits h is initiated to the value of high order digit position of an m-digit window p and t0 are computed in O(m+m) time The loop of line 9 takes Q((n-m+1)m) time The loop 6-8 takes O(m) time The overall running time is O((n-m)m) qqqOqD6DH , 0޽h ? ̙330  @0(  @x @ c $iY H `}  T  x @ c $jY H ` Y  H @ 0޽h ? 380___PPT10.*$  0P$(  Pr P S xY H `}  Y  r P S nY H ` Y  H P 0޽h ? 3380___PPT10.?PoOrXC&/(,m4W:v>IO]hA  DIYm 1o ? q Oh+'0| hp    String Matching Algorithms  Mohan Kumaroha Mohan Kumar38aMicrosoft PowerPointith@|@`,`}@ 0GT g  .  -- @ !--'@Arial-. *2 Kumar String matching      ."System-@Arial-.  2 1 .-@Arial-. .2 \String Matching Algorithms)%%3""%&%*&%%6".-@Arial-. 2 Topics .-@Wingdings-.  2 q.-@Arial-. !2 "Basics of Stringst    .-@Wingdings-.  2 q.-@Arial-. 2 "Brutes  .-@Arial-.  2 x- .-@Arial-. %2 force String Matcher        .-@Wingdings-.  2 1q.-@Arial-. 2 1"Rabins .-@Arial-.  2 1~- .-@Arial-. 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Divide and Conquer principle Divide a problem into smaller independent subproblems Solve these Sub problems individually Conquer: Solve the main problem by combining the solutions of sub problemsnP7P(PKP  X A uQuick Sort Algorithm fGiven an array of n values, randomly pick an element (pivot) in the array to partition. Then compare the rest of the elements to this value. If they are greater, put them to the  right of the partition element If they are less, put them to the  left of the partition element. When done with the partition, the position of the pivot element is fixed. Sort the left partition, using Quick Sort (Recursive!) Sort the right part of the array, using Quick Sort. (Recursive!)8 " 6^PQuick Sort Partition Algorithm$fint partition1( int *a, int low, int high ) { int left, right, pivot_item; pivot_item = a[low]; left = low; right = high; while ( left < right ) { /* Move left while item < pivot */ while( a[left] <= pivot_item ) left++; /* Move right while item > pivot */ while( a[right] >= pivot_item ) right--; if ( left < right ) SWAP(a,left,right); } /* right is final position for the pivot */ SWAP(a, low, right); return right; } 4PPZ  "Q U I C K S O R T contd..* ffA simple AnimationU 0%Q U I C K S O R T Analysis: f$f6fFocus of analysis: Selecting Pivot element / partition Quick Sort Running Time: T(n) = n-1 + T(i-1) + T(n-i) --- (1) BESTCASE: Split in the middle, pivot = n/2 T(n) = 2T(n/2) + O(n), Solving recursive function gives O(nlogn), the ideal case of QuickSort. WORST CASE: Pivot = last or first element of array T(n) = n-1 + T(0) + T(n-1) from(1) sub i = 1 or n T(n) = O(n2) AVERAGE CASE : Split at random position T(n) = 2(T(0)+. . .+T(n-1))/n + O(n) 7PPkPoPZPPOP$   :  Z     ?, ' @Improving Worst case Choose random partitioning element Randomized QuickSort by Hoare 1960 Randomized QuickSort keeps running time independednt of Input Other ways  Median of three elements as pivot Randomly pick three elements in the array to be sorted Choose the middle value of these three elements to be the partition Complexity  Less if size of array to be sorted is large. Gain: Good partition element PPP1PP/>E   gV 3 way Radix Quick Sort( (f cS3 way partitioningf dTDutch National Flag Problem$fProblem: Re-arrange an array of elements in to top, middle and bottom An Elegant solution to Dutch national flag problem. Partition elements into 4 parts:  no larger elements to left of m  no smaller elements to right of m  equal elements to left of p  equal elements to right of q Then, swap equal keys to center of array.XZZ,Z+fU 3 way Radix Quick Sort Algorithm!!$fprivate static void quicksortX(String a[], int lo, int hi, int d) { if (hi - lo <= 0) return; int i = lo-1, j = hi, p = lo-1, q = hi; char v = a[hi].charAt(d); while (i < j) { while (a[++i].charAt(d) < v) ; while (v < a[--j].charAt(d)) if (j == lo) break; if (i > j) break; exch(a, i, j); if (a[i].charAt(d) == v) { p++; exch(a, p, i); } if (a[j].charAt(d) == v) { q--; exch(a, j, q); } } if (p == q) { if (v != '\0') quicksortX(a, lo, hi, d+1); return; } if (a[i].charAt(d) < v) i++; for (int k = lo; k <= p; k++, j--) exch(a, k, j); for (int k = hi; k >= q; k--, i++) exch(a, k, i); quicksortX(a, lo, j, d); if ((i == hi) && (a[i].charAt(d) == v)) i++; if (v != '\0') quicksortX(a, j+1, i-1, d+1); quicksortX(a, i, hi, d); }"P    4#+, % &  hWSome sample Functions(fRecords with equal keys together. 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S  0\ "^ ` \ X*  0T\ "^  \ Z*  0<\ "^ ` \ Z*H  0޽h ? 3380___PPT10.pVh Default DesignB  ` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> ph(    60c " `} c T Click to edit Master title style! !$  03c " ` c RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0:c "^ ` c Z*"  0=c "^  c \*"  0Bc "^ ` c \*"H  0޽h ? 3380___PPT10.,h!  1_Default Design@ 0 zrP (    0Qc P   c P*    0$Wc    c R*  d  c $ ?  c  0DRc  0 c RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  6]c _P  c P*    60cc _  c R*  H  0޽h ? 3380___PPT10.p #0:@ QI0(    <_\`]  USTUDY OF SORTING ALGORITHMS f  h WԔ?HARIKRISHNA MALAYAPPAN KUMAR KRIS RAMANUJAM Times New Roman"`P0   <d\p W  Z$THE UNIVERSITY OF TEXAS AT ARLINGTON% %H  0޽h ? 33___PPT10i.o 0H+D=' *= @B +s@ @p(  p p 0qc|}   First intensively studied computer science problem. Developed during 1960s when large scale commercial data processing System was automated on a large scale Transfer of data between slow storage (tape or disk) and main memory was a major issue. Most files does not fit inside the memory Main Memory was limited (in 100 K bytes). Files to be sorted were in large magnitude. Algorithm to sort these files was a major focus among companies.  5  D + @  +  Y  B  4oW*,u p <0wcY* >Sorting  Historic Introduction. ffH p 0޽h ? 3333___PPT10i.u)Q+D=' *= @B +@ Pt&(  t& t 0cP   Practical use since often used Working with large sets of data in computers is facilitated when the data is sorted. Different perspective towards the same problem e.g. How to improve the worst case time of the Quick Sort? Minimum amount of work specified by the algorithms to get the optimal solution to the problem. How much extra space does the algorithm use to optimize the memory usage If the amount of extra space is constant with respect to the input size, the algorithm is said to work IN PLACE.  h  0 < @ & D  N (  !\0;fD u t <c T Motivation.  ffH t 0޽h ? 3333___PPT10i.v) 3+D=' *= @B +@ `E(  ,  0> n i.e. efficient for array with equal keys 3  $  Q 9   f$  8X G <1, and we perform the following three steps in sequence: Sort the left half of the the array. Sort the right half of the the array. Merge the now-sorted left and right halves. Z  |  %  |  >&'1H  0޽h ? 3333 @ 0=(  r  S ^z    z   6lbz %WWorst-case number of comparisons T(n) = T(n/2) + T(n/2) + n, if n>1. First term - number of comparisons used to sort the left half of the array Second term - number of comparisons used to sort the left half of the array Third term - an upper bound on the number of comparisons used to merge two sorted arrays G 0 " M D \H  0޽h ? 3333r @ "@(  r  S tz    z   0vz@pJ LSuppose we want to sort 16 elements, T(16) = ? T(16) = 2T(8) + 16 T(8) = 2T(4) + 8 T(4) = 2T(2) + 4 T(2) = 2T(1) + 2 T(1) = 0 i.e. T(1) = 0 T(2) = 2T(1) + 2 = 0 + 2 T(4) = 2T(2) + 4 = 4 + 4 = 8 T(8) = 2T(4) + 8 = 16 + 8 = 24 T(16) = 2T(8) + 16 = 48 + 16 = 64 So MergeSort requires at most 64 comparisons to sort 16 elements. M M  7H  0޽h ? 3333@  |(  |P | <z  Good running time for most data sets. Based on divide and conquer approach. Chops partition half way by dividing (left + right)/2 Successive pair of elements are used to combine when we merge This algorithm works in two sorted input lists of unequal size. Merging takes n/2 comparisons. No more than (n  1) comparisons., N O | BzP9 $M E R G E S O R T O V E R V I E WR%  fff$fH | 0޽h ? 3F433___PPT10i.pӫ+D=' *= @B +9@ PH0((  ( ( 0zqT l Merge Sort uses the divide and conquer approach. Divides the element into two sub lists n/2. If there are n elements, two sub arrays are created such that Left Sub Array [ 1& & .r] Right Sub Array [r +1, n] Both the sub Arrays are sorted accordingly. Once the Arrays are sorted they are combined into a single array for A[1& .n]. (Conquer Approach) Sorting is done in the Merge itself. Conquering is done recursively using the Merge Sort.   8 o " \  _x,e[ ( <z hM E R G E S O R T ( C O N T).  f fH ( 0޽h ? 3333___PPT10i.'®+D=' *= @B + C @m(  X  0P   <zj]Q [)First floor (First + Last) /2 Last* *LB  c $D LB  c $D@ @ RB  s *D `RB  s *D `  6PzP >     < zF DSorted Sorted    68z`   >  RB   s *D@` RB  @ s *D p`    <мzz   9Sorted    <|zn PSort Recursively by Merge Sort   <@z f  7Merge   <zpp.W ^(Diagrammatic representation - Merge Sort) )fH  0޽h ? 3333___PPT10i.-O+D=' *= @B +D@ [SP*0,(  , , <z. I M E R G E E X A M P L E O N T W O S O R T E D S U B A R R A YS4J Hff!l @  , #"0000  , <pz?\@  G2 @` , <z?@\ G1 @` , <Tz?@ G0 @``B  , 0o ?@ZB  , s *1 ?@ZB  , s *1 ?\@\`B  , 0o ? @ `B  , 0o ? `B , 0o ?@@ 'r @  ,# #"000  , <z?\@  G5 @` , <z?@\ G4 @` , <}?@ G3 @``B , 0o ?@ZB , s *1 ?@ZB , s *1 ?\@\`B , 0o ? @ `B , 0o ? `B , 0o ?@@  , < }* LL E F T S U B A R R A Y  , <`}g NR I G H T S U B A R R A Y x pp  ., #"*HHHHHH   $, <d}? p  G5 @` #, <}?   G4 @` ", <H)}?p  G3 @` !, <1}?p G2 @`  , <9}? G1 @` , <h;}?p G0 @``B %, 0o ?ppZB &, s *1 ?ZB ', s *1 ?ZB (, s *1 ?ppZB ), s *1 ?  ZB *, s *1 ?  `B +, 0o ?p p `B ,, 0o ?pp `B -, 0o ?pp o /, <E} @I A comparison is made between the first element of the left sub array And the right sub array. Which ever is greater goes into the first place of The main Array. The appropriate index of the array is increased.  0, <F}`@ G EM A I N A R R A Y H , 0޽h ? 3333___PPT10i.'Y>++D=' *= @B +<@ `-50(  0 0 0 W}&p ,$  0 r&MERGE EXAMPLE ON TWO SORTED SUB ARRAYS0' &ffX   ,0 #"000 ,$D  0 0 <\z?\  H40 @` 0 <f}?\ H35 @`  0 <8h}? H30 @``B  0 0o ?ZB  0 s *1 ?ZB  0 s *1 ?\\`B  0 0o ?  `B 0 0o ? `B 0 0o ? X P  30 #"`00P ,$D  0 0 <Lw}?Pv  H30 @` 0 <q}?P0v H20 @` 0 <0}?P0 H10 @``B 0 0o ?PZB 0 s *1 ?P00ZB 0 s *1 ?Pvv`B 0 0o ?P  `B 0 0o ?PP `B 0 0o ?  pp  0# #"*HHHHHH  ,$D  0 0 <H}? p  H40 @` 0 <}?   H35 @` 0 <}?p  H30 @` 0 <}?p H30 @` 0 <}? H20 @`  0 <}?p H10 @``B !0 0o ?ppZB "0 s *1 ?ZB #0 s *1 ?ZB $0 s *1 ?ppZB %0 s *1 ?  ZB &0 s *1 ?  `B '0 0o ?p p `B (0 0o ?pp `B )0 0o ?pp  *0 0(} ,$  0 90 1 2  -0 0}0k ,$  0 93 4 5  .0 0 }p< ,$  0 B0 1 2 3 4 5  10 <0}`D G,$   0 <  f 20 0} @a,$   0 ~Left[0] is compared with Right[0]. R[0] is greater than L[0] R[0] is placed in the MainArray at 0. R index is increased by 1. T " 40 <}*,$   0 LL E F T S U B A R R A Y  50 <}g,$   0 NR I G H T S U B A R R A Y H 0 0޽h ? 3333_ W ___PPT107 .'AI!+ ҚCDK' *= @B D' = @BA?%,( < +O%,( < +D=' (=%(%(D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-m6Bbox(in)*<3<*0D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,0%(D' =-m6Bbox(in)*<3<*,0D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*30%(D' =-m6Bbox(in)*<3<*30D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-m6Bbox(in)*<3<*0D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<**0%(D' =-m6Bbox(in)*<3<**0D' =%( D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*-0%(D' =-m6Bbox(in)*<3<*-0D' =%( D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*.0%(D' =-m6Bbox(in)*<3<*.0D' =%( Dh' =K@BBBBPB0B%(/%(!D' =1:Bvisible*o3>+B#style.visibility<*10%(D' =-m6Bbox(in)*<3<*10D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*20%(D' =-m6Bbox(in)*<3<*20D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*40%(D' =-m6Bbox(in)*<3<*40D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*50%(D' =-m6Bbox(in)*<3<*50++0+0} ++0+*0} ++0+-0} ++0+.0} ++0+10} ++0+20} ++0+40} ++0+50} +B VVp1V(   c # #"   c,$D 0  <~?Pc L64  @`  < ~? 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L14  @``B D 0o ?  `B E 0o ?  `B F 0o ?  `B G 0o ?  7  @  H# #" @ @,$D   0 I <~?` @  L22  @` J <~? @`  K8  @``B K 0o ? @ @`B L 0o ?  `B M 0o ? @ ZB N s *1 ?` @` `B O 0o ? @  @ c P# #" @ ,$D  0 Q <p~?@ c L64  @``B R 0o ?@ `B S 0o ?@c c`B T 0o ?@@c`B U 0o ?  c   S  V# #" p S,$D  0 W <~? S  K8  @``B X 0o ? `B Y 0o ? S S `B Z 0o ? S `B [ 0o ? S  P pS \# #" P pS,$D  0 ] <~?P pS L22  @``B ^ 0o ?P p`B _ 0o ?P SpS`B ` 0o ?P P S`B a 0o ?ppSB b@ s *D0 ,$D 0B c s *D0 ,$D 0  d# #" @0,$D'  0 e <8~? J64 @` f <~?  J22 @` g <~?   J20 @` h <D?  J14 @` i < ? J13 @` j <? I8 @``B k 0o ?`B l 0o ?`B m 0o ?ZB n s *1 ?ZB o s *1 ?ZB p s *1 ?  ZB q s *1 ?  ZB r s *1 ?`B s 0o ?b    t# #" | `P ,$D#  0 u <?   L20  @` v <&?P   L14  @` w </? P  L13  @``B x 0o ?  `B y 0o ?  `B z 0o ? ZB { s *1 ?P P ZB | s *1 ?  `B } 0o ?  a     ~# #" _  ,$D$  0  <:?   L64  @`  <C?@    L22  @`  <M?  @  K8  @``B  0o ?   `B  0o ?  `B  0o ?  ZB  s *1 ?@  @ ZB  s *1 ?  `B  0o ?  8  Z -  # #"  PP ,$D  0  <$X?xZ -  L20  @`  <`? Z x-  L14  @``B  0o ? Z Z `B  0o ? - - `B  0o ? Z - ZB  s *1 ?xZ x- `B  0o ?Z - 7  p  # #"   ,$D  0  <k? p  L22  @`  <t?  K8  @``B  0o ? p `B  0o ? p `B  0o ?  ZB  s *1 ? `B  0o ?p p B  s *D ,$D 0B @ s *Dp ,$D 0B  s *D P ,$D  0B @ s *D001 ,$D  0B  s *D `P@,$D& 0B  s *Dp 0 ,$D 0B @ s *D@ ,$D 0B @ s *D ,$D  0B @ s *D0 ,$D   0B @ s *D @,$D% 0H  0޽h ? 33___PPT10.unR+[D' *= @B DO' = @BA?%,( < +O%,( < +D' (=%(%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =%( DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-6B'blinds(horizontal)*<3<*(D' =%( DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-6B'blinds(horizontal)*<3<*)D' =%( DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-6B'blinds(horizontal)*<3<**D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =-6B'blinds(horizontal)*<3<*+D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4D' =%(|D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*HD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*HD' =%(pD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*PD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*PD' =%(dDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =-6B'blinds(horizontal)*<3<*;D' =%(XDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*:%(D' =-6B'blinds(horizontal)*<3<*:D' =%(LDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*b%(D' =-6B'blinds(horizontal)*<3<*bD' =%(@DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*c%(D' =-6B'blinds(horizontal)*<3<*cD' =%(4!D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*<D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*<D' =%((#D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*B%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*BD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*BD' =%(%D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*V%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*VD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*VD' =%('D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*\D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*\D' =%()DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(,DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(.DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(0D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(2D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(4D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(6D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(8D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(:D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(<DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(>DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(t@D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*tD' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*tD' =%(hBD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*~%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*~D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*~D' =%(\DDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(PFDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DHD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*dD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*d+ @   8m (  8 8 B0{# `(M E R G E S O R T A N A L Y S I S) )f  8 0|   MDepth # sequences size 0 1 n 1 2 n/2 2 22 n/22 BN C   8 <$ F  c 3 23 n/238    8 <ܾj  Q y i 2i n/2i B   X2  8 0 RB  8@ s *D 0RB  8 s *D`X2  8 0 0 X2  8 0@0 RB 8@ s *D0 RB 8 s *D0 RB 8@ s *D0 @ RB 8 s *D0 ` X2 8 0 P X2 8 0 P X2 8 0 0P X2 8 0 `@P RB 8@ s *DP RB 8 s *DP ` RB 8@ s *DP RB 8 s *DP @ RB 8@ s *DP  RB 8 s *DP @ RB 8@ s *DP ` RB 8 s *DP  H 8 0޽h ? 3333___PPT10i.'@o+D=' *= @B +7@ NF&(  r # C JA "img2211displaymath70467`z $ C DA img2212eqnarray44644"`P` % BP^ o=Source: http://www.brpreiss.com/books/opus4/html/page513.html> > & 00P a)Calculation of Running Time Of Merge Sort* *fH  0޽h ? 3333___PPT10i.C"-+D=' *= @B +$@ ;34(  4 4 6 Vp Operations in O(n lg n) time. No best  case and worst case as Quick Sort where the worst case is O(n2) Needs a duplicate copy of the array being sorted. Merge Sort is used mainly for small size of n. If n is large it is very impractical to apply the algorithm. Merge Sort is fast for small n.  d      L 2# 4 <#  c+M E R G E S O R T E F F I C I E N C Y, ,fH 4 0޽h ? 3333___PPT10i.'@Z\+D=' *= @B +0"@ G!?!i< (  < < BD  `(M E R G E S O R T A N A L Y S I S) )fx 2< <A o??@0  oE   i< #">2  N _< B?O VNF___PPT9(  v33.21928< (" dcC @`Q ^< B? OVNF___PPT9(  y 10000000000< (" d cC @`N ]< B ?O  VNF___PPT9(  v29.89735< (" dcC @`P \< B ? O VNF___PPT9(  x 1000000000< (" d cC @`N [< B`$?O  VNF___PPT9(  v26.57542< (" dcC @`O Z< B.? O VNF___PPT9(  w 100000000< (" d cC @`M Y< Bp8?O  VNF___PPT9(  u23.2535<(" dcC @`N X< Bl;? O VNF___PPT9(  v10000000< (" dcC @`N W< B(" dcC @`I J< B? OVNF___PPT9(  qn>(" dcC @`HB `< C  HB a< C  HB b< C   HB c< C H < 0޽h ? 3333___PPT10i.'BW +D=' *= @B +?C >>J%R>(  $> p- % #"VJp- $ Bނ? Zp- m 250000.00000 ( CC @` $ Bx? Z - k 4482.89210 ( CC @` $ Bp?Z - h8.96580(CC @` $ B?Z- e5000(CC @` $ B? pZ m 160000.00000 ( CC @` $ B? Z k 3457.54250 ( CC @` $ B? Z h8.64390(CC @` $ B(? Z U400 (C @` $ B2? p  l 90000.00000 ( CC @` $ Bl<?  k 2468.64560 ( CC @` $ BE?  h8.22880(CC @` $ B$O?   U300 (C @` $ BQ? p  l 40000.00000 ( CC @` $ Bc?  k 1528.77120 ( CC @` $ Bm?  h7.64390(CC @` $ B8g?   e2000(CC @` $ Bp?  p  \ 36100.0000 ( C @` $ B艄?   [ 1438.2726 ( C @` $ BD{?  X7.5699 (C @` $ B4?   U190 (C @` $ Bܤ? ; p  \ 32400.0000 ( C @` $ B䦄? ;   [ 1348.5336 ( C @` $ B(?;   X7.4919 (C @` $ B|?;   U180 (C @` $ B Ʉ? h p;  \ 28900.0000 ( C @` $ B,҄? h ;  [ 1259.5965 ( C @` $ Bdۄ?h ;  X7.4094 (C @` $ Bx?h ;  U170 (C @` $ BՄ? ph  \ 25600.0000 ( C @` $ B4?  h  [ 1171.5085 ( C @` $ BD? h  X7.3219 (C @` $ B ?h  U160 (C @` $ B? p l 22500.00000 ( CC @` $ B?   k 1084.32280 ( CC @` $ Bl&?  h7.22880(CC @` $ B@0? e1500(CC @` $ B:? p \ 19600.0000 ( C @` $ B4C?   Z998.0996 ( C @` $ BXL?  X7.1293 (C @` $ BT? U140 (C @` $ B^? p \ 16900.0000 ( C @` $ B`?   Z912.9078 ( C @` $ Bp?  X7.0224 (C @` $ By? U130 (C @` $ B{? Ip \ 14400.0000 ( C @` $ B(? I  Z828.8269 ( C @` $ Bt?I  X6.9069 (C @` $ BP?I U120 (C @` $ B? vpI \ 12100.0000 ( C @` $ B? v I Z745.9496 ( C @` $ BĹ?v I X6.7814 (C @` $ B(…?vI U110 (C @` $ B̅? pv l 10000.00000 ( CC @` $ Bօ?  v j664.38560 (CC @` $ B? v h6.64390(CC @` $ B?v e1000(CC @` $ B? p k 2500.00000 ( CC @` $ BD?   j282.19280 (CC @` $ B ?  h5.64390(CC @` $ Bt? d500(CC @` $ B? p Z100.0000 ( C @` $ B%?   Y33.2193 (C @` $ B/?  X3.3219 (C @` $ B8? T10 (C @` $ B4C? p n2 Quick Sort (Worst Case)R(CKCC @` $ B,M?  in log n0(CC @` $ BLV?  glog n0(CC @` $ B_? cn0(CC @`HB $ C pHB $ C -p-HB $ C -HB % C pp-H $ 0޽h ? 3333___PPT10i.D"@+D=' *= @B +@ /' (      0  f.Q U I C K S O R T V S M E R G E S O R T/ /f   <  M Quick Sort does not require an extra array to work whereas Merge Sort needs an extra array to merge the two sorted sub lists into the main array. Merge Sort has better worst case behavior than Quick Sort A general sorting algorithm cannot do better than a average case of O (n log n). Merge Sort is a sorting algorithm which is not in-place when dealing with arrays. = _   F  =_:WVH   0޽h ? 333380___PPT10.f.pj:C QI(    0 IH E A P S O R T  fH  0޽h ? 33___PPT10i. RQ+D=' *= @B +n@ }(    <  Heap Sort is one of the best general-purpose algorithms Part of Selection Sort Family Very good running time performance on randomly ordered arrays. Worst case is same as the average case performance. Needs only a fixed amount of extra storage space. In-Place algorithm. vY   9   BܢN 9 HH E A P S O R T  fH  0޽h ? 3F433___PPT10i.`+D=' *= @B +h@ wD(  D D <@P  \ Heap Sort basically works with an array. There are two types of heaps. Min Heaps Max Heaps. Smallest element is Largest Element is at the root. At the root.  L b +! D D <   Every new element taken from the array has to be inserted into the correct place. Once inserted into an heap we got to address the locality of the element. TA  C   D BN = HH E A P S O R T  fH D 0޽h ? 3333___PPT10i.'+D=' *= @B +\fC ZR(     <|,$ 0 l:I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5 (Max Heap); ;2  6Ɔ `,$D  0 >    <ɆZ6, A,$  0 411 2  6$ΆP ,$D 0 >  2  6цP ,$D  0 >  2  6Ԇ  p,$D   0 >  2  6ن @p,$D  0 >  2   6݆P 0 ,$D  0 >  2   6HP,$D  0 >  2   6P 0 ,$D  0 >  B  @ s *D0@,$D   0B   s *D P,$D   0B @ s *D@P,$D  0B  s *DP ,$D   0B @ s *Dpp P,$D  0B  s *Dp pP,$D  0  <t <    <g,$ 0 420   <@j  Q,$ 0 415   <$:!,$  0 430   <g,$  0 425   <D * ,$  0 440   <jfQ,$  0 456 2  6P P ,$D  0 >    <   ,$  0 33   <j Q ,$  0 35 B @ s *DppP ,$D  0B  s *DpP ,$D  0  <@ . h6B U I L D H E A P Heap from a unordered input array7 7  <  &  2Steps: Build_heap(A) heap_size(A) = |A| for i = (A/2) down to 1 Heapify (A, i) log n Running time is " n/ 2i = O(n) i = 1,  ENH  0޽h ? fp=LL___PPT10L.8t+D.H' *= @B DG' = @BA?%,( < +O%,( < +D G' (=%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%( DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(|D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(pD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(dD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(XD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(LD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(@D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4!D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%((#D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(%D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%('D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%()D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(,D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(.D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(0D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(2D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*+(+0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ +jC   x(    <R,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ /2  6X `,$D  0 >    <[Z6, A,$  0 411 2  68`P ,$D 0 >  2  6cP ,$D  0 >  2  6f  p,$D   0 >  2  6,k @p,$D  0 >  2   6TP 0 ,$D  0 >  2   6rP,$D  0 >  2   6vP 0 ,$D  0 >  B  @ s *D0@,$D   0B   s *D P,$D   0B @ s *D@P,$D  0B  s *DP ,$D   0B @ s *Dpp P,$D  0B  s *Dp pP,$D  0  <|t <    <g,$ 0 420   <j  Q,$ 0 415   <؆:!,$  0 430   <g,$  0 425   <쎇 * ,$  0 440   <ؒjfQ,$  0 456 2  6P P ,$D  0 >    <  ,$  0 33   <j Q ,$  0 35 B @ s *DppP ,$D  0B  s *DpP ,$D  0  <l  ]H E A P I F Y Checks for heap property. If the heap property is violated Restores the heap^ ^B  0D`,$D  0   <䢇 p { {IComparing 11, 20 and 15 20 is greater than 11 and 15. Move 20 to root.J JH  0޽h ? fp=QQ___PPT10Q.+yvXD M' *= @B DL' = @BA?%,( < +O%,( < +DK' (=%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%( DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(|D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(pD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(dD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(XD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(LD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(@D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4!D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%((#D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(%D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%('D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%()D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(,D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(.D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(0D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(2D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4DD' =A@BBBB0B%(D' =-u6Bdiamond(in)*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(+`+0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+  +jC 0 w(    <,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ /2  68 `,$D  0 >    <Z6Z A,$  0 420 2  6xP ,$D 0 >  2  68P ,$D  0 >  2  6  p,$D   0 >  2  6 @p,$D  0 >  2   6xP 0 ,$D  0 >  2   6H P,$D  0 >  2   6P 0 ,$D  0 >  B  @ s *D0@,$D   0B   s *D P,$D   0B @ s *D@P,$D  0B  s *DP ,$D   0B @ s *Dpp P,$D  0B  s *Dp pP,$D  0  < t <    <g,$ 0 411   <\j  Q,$ 0 415   <:!,$  0 430   <!g,$  0 425   <|( * ,$  0 440   <"jfQ,$  0 456 2  60P P ,$D  0 >    <\3  ,$  0 33   <h7j Q ,$  0 35 B @ s *DppP ,$D  0B  s *DpP ,$D  0  <>  ]H E A P I F Y Checks for heap property. If the heap property is violated Restores the heap^ ^B  0D`,$D  0   <?  zHComparing 11, 30 and 25 30 is greater than 11 and 25. Swap 30 with 11I IH  0޽h ? fp=QQ___PPT10Q./؜+yvXD M' *= @B DL' = @BA?%,( < +O%,( < +DK' (=%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%( DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(|D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(pD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(dD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(XD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(LD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(@D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4!D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%((#D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(%D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%('D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%()D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(,D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(.D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(0D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(2D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4DD' =A@BBBB0B%(D' =-u6Bdiamond(in)*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(+`+0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+  + kC @ (    <,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ /2  6 `,$D  0 >    <Z6Z A,$  0 420 2  6莊P ,$D 0 >  2  6ԑP ,$D  0 >  2  6ؔ  p,$D   0 >  2  6 @p,$D  0 >  2   6P 0 ,$D  0 >  2   6P,$D  0 >  2   6(P 0 ,$D  0 >  B  @ s *D0@,$D   0B   s *D P,$D   0B @ s *D@P,$D  0B  s *DP ,$D   0B @ s *Dpp P,$D  0B  s *Dp pP,$D  0  <4t <    <<g,$ 0 430   <j  Q,$ 0 415   <\:!,$  0 411   <4g,$  0 425   <p * ,$  0 440   <\jfQ,$  0 456 2  6ŊP P ,$D  0 >    <dȊ  ,$  0 33   <@̊j Q ,$  0 35 B @ s *DppP ,$D  0B  s *DpP ,$D  0  <ʊ  ]H E A P I F Y Checks for heap property. If the heap property is violated Restores the heap^ ^B  0D`,$D  0   <hъ  TComparing 30, 20 and 15 30 is greater than 20 and 15. Move 30 in the place of 20.U UH  0޽h ? fp=QQ___PPT10Q.+yvXD M' *= @B DL' = @BA?%,( < +O%,( < +DK' (=%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%( DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(|D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(pD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(dD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(XD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(LD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(@D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4!D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%((#D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(%D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%('D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%()D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(,D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(.D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(0D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(2D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4DD' =A@BBBB0B%(D' =-u6Bdiamond(in)*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(+`+0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+  +hC P O(    <,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ /2  6 `,$D  0 >    <!Z6Z A,$  0 430 2  6P ,$D 0 >  2  6D)P ,$D  0 >  2  6,  p,$D   0 >  2  6/ @p,$D  0 >  2   6l3P 0 ,$D  0 >  2   67P,$D  0 >  2   6t:P 0 ,$D  0 >  B  @ s *D0@,$D   0B   s *D P,$D   0B @ s *D@P,$D  0B  s *DP ,$D   0B @ s *Dpp P,$D  0B  s *Dp pP,$D  0  <At <    <Dg,$ 0 420   <Hj  Q,$ 0 415   <@:!,$  0 411   <LOg,$  0 425   <$S * ,$  0 440   <VjfQ,$  0 456 2  68[P P ,$D  0 >    <d^  ,$  0 33   <aj Q ,$  0 35 B @ s *DppP ,$D  0B  s *DpP ,$D  0  <e  ]H E A P I F Y Checks for heap property. If the heap property is violated Restores the heap^ ^  <xj @ m TComparing 11, 20 and 25 25 is greater than 20 and 11. Move 25 in the place of 20.U UXB   0D H  0޽h ? fp=O~O___PPT10^O.ER+DJ' *= @B DJ' = @BA?%,( < +O%,( < +DI' (=%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%( DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(|D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(pD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(dD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(XD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(LD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(@D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4!D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%((#D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(%D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%('D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%()D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(,D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(.D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(0D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(2D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4DD' =A@BBBB0B%(D' =-u6Bdiamond(in)*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(+`+0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ +hC ` O(    <x,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ /2  6 `,$D  0 >    <0Z6Z A,$  0 430 2  6TP ,$D 0 >  2  6P ,$D  0 >  2  6\  p,$D   0 >  2  6,ƍ @p,$D  0 >  2   6ȍP 0 ,$D  0 >  2   6l̍P,$D  0 >  2   6эP 0 ,$D  0 >  B  @ s *D0@,$D   0B   s *D ,$D   0B @ s *D@P,$D  0B  s *DP ,$D   0B @ s *Dpp P,$D  0B  s *Dp pP,$D  0  <؍t <    <ۍg,$ 0 425   <Xj  Q,$ 0 415   <D:!,$  0 411   <g,$  0 420   < * ,$  0 440   <0jfQ,$  0 456 2  6P P ,$D  0 >    <  ,$  0 33   <lj Q ,$  0 35 B @ s *DppP ,$D  0B  s *DpP ,$D  0  <D  ]H E A P I F Y Checks for heap property. If the heap property is violated Restores the heap^ ^  <@  TComparing 15, 40 and 56 56 is greater than 15 and 40. Move 56 in the place of 15.U UXB   0D pH  0޽h ? fp=O~O___PPT10^O.+DJ' *= @B DJ' = @BA?%,( < +O%,( < +DI' (=%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%( DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(|D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(pD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(dD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(XD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(LD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(@D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4!D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%((#D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(%D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%('D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%()D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(,D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(.D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(0D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(2D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4DD' =A@BBBB0B%(D' =-u6Bdiamond(in)*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(+`+0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ +hC p O(    <4D,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ /2  6(J `,$D  0 >    <0MZ6Z A,$  0 430 2  6QP ,$D 0 >  2  6UP ,$D  0 >  2  6XX  p,$D   0 >  2  6\ @p,$D  0 >  2   6\aP 0 ,$D  0 >  2   6dP,$D  0 >  2   6LhP 0 ,$D  0 >  B  @ s *D0@,$D   0B   s *D ,$D   0B @ s *D@P,$D  0B  s *DP ,$D   0B @ s *Dpp P,$D  0B  s *Dp pP,$D  0  <0nt <    <8qg,$ 0 425   <tj  Q,$ 0 456   <Xx:!,$  0 411   <0|g,$  0 420   <l * ,$  0 440   <XjfsQ,$  0 415 2  6 P P ,$D  0 >    <`  ,$  0 33   <<j Q ,$  0 35 B @ s *DppP ,$D  0B  s *DpP ,$D  0  <썏  ]H E A P I F Y Checks for heap property. If the heap property is violated Restores the heap^ ^  <d@ p  TComparing 56, 25 and 30 56 is greater than 25 and 30. Move 56 in the place of 30.U UXB   0DP p H  0޽h ? fp=O~O___PPT10^O.+DJ' *= @B DJ' = @BA?%,( < +O%,( < +DI' (=%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%( DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(|D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(pD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(dD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(XD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(LD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(@D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4!D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%((#D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(%D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%('D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%()D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(,D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(.D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(0D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(2D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4DD' =A@BBBB0B%(D' =-u6Bdiamond(in)*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(+`+0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ +hC  O(    <ُ,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ /2  6,ޏ `,$D  0 >    <Z6Z A,$  0 456 2  6\P ,$D 0 >  2  6P ,$D  0 >  2  6t  p,$D   0 >  2  6  @p,$D  0 >  2   6P 0 ,$D  0 >  2   6P,$D  0 >  2   6DP 0 ,$D  0 >  B  @ s *D0@,$D   0B   s *D ,$D   0B @ s *D@P,$D  0B  s *DP ,$D   0B @ s *Dpp P,$D  0B  s *Dp pP,$D  0  <0t <    <8 g,$ 0 425   <j  Q,$ 0 430   <$:!,$  0 411   <Hg,$  0 420   <L * ,$  0 440   <tjfsQ,$  0 415 2  6"P P ,$D  0 >    <%  ,$  0 33   <(j Q ,$  0 35 B @ s *DppP ,$D  0B  s *DpP ,$D  0  <.  ]H E A P I F Y Checks for heap property. If the heap property is violated Restores the heap^ ^  <01@ p  TComparing 30, 40 and 15 40 is greater than 30 and 15. Move 40 in the place of 30.U UXB  @ 0Dp p H  0޽h ? fp=O~O___PPT10^O.{Q+DJ' *= @B DJ' = @BA?%,( < +O%,( < +DI' (=%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%( DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(|D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(pD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(dD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(XD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(LD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(@D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4!D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%((#D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(%D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%('D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%()D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(,D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(.D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(0D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(2D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4DD' =A@BBBB0B%(D' =-u6Bdiamond(in)*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(+`+0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ +lC  (    <q,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ /2  6w `,$D  0 >    <zZ6Z A,$  0 456 2  6~P ,$D 0 >  2  6䁒P ,$D  0 >  2  6脒  p,$D   0 >  2  6 @p,$D  0 >  2   6P 0 ,$D  0 >  2   6P,$D  0 >  2   68P 0 ,$D  0 >  B  @ s *D0@,$D   0B   s *D ,$D   0B @ s *D@P,$D  0B  s *DP ,$D   0B @ s *Dpp P,$D  0B  s *Dp pP,$D  0  <Dt <    <Lg,$ 0 425   <j  Q,$ 0 440   <l:!,$  0 411   <Dg,$  0 420   < * ,$  0 430   <ljfsQ,$  0 415 2  6 P P ,$D  0 >    <t  ,$  0 33   <Pj Q ,$  0 35 B @ s *DppP ,$D  0B  s *DpP ,$D  0  <  ]H E A P I F Y Checks for heap property. If the heap property is violated Restores the heap^ ^'  <<Œ0 Uq,$ 0 UH e a p I f y a l g o r i t h m s D o n e f o r t h e g i v e n I n p u t V VXB  @ 0Dp @ H  0޽h ? fp=RR___PPT10R.@p+cӰDM' *= @B DM' = @BA?%,( < +O%,( < +DI' (=%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%( DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(|D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(pD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(dD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(XD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(LD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(@D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4!D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%((#D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(%D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%('D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%()D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(,D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(.D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(0D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(2D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4DD' =A@BBBB0B%(D' =-u6Bdiamond(in)*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' (=%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ +mC +#")(    <0 @] |,H e a p I n s e r t i o n V a l u e = 304- ff  <,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ /2  6 `,$D  0 >    <Z6Z A,$  0 456 2  6 P ,$D 0 >  2   6P ,$D  0 >  2   6"  p,$D   0 >  2   6& @p,$D  0 >  2   60*P 0 ,$D  0 >  2   6.P,$D  0 >  2  6p1P 0 ,$D  0 >  B @ s *D0@,$D   0B  s *D ,$D   0B @ s *D@P,$D  0B  s *DP ,$D   0B @ s *Dpp P,$D  0B  s *Dp pP,$D  0  <d8t <    <P;g,$ 0 425   <p?j  Q,$ 0 440   <6:!,$  0 411   <Fg,$  0 420   <I * ,$  0 430   <MjfsQ,$  0 415 2  6RP P ,$D  0 >    <0U  ,$  0 33   <Yj Q ,$  0 35 B @ s *DppP ,$D  0B   s *DpP ,$D  0XB #@ 0Dp @ ? $ 0] h  Add 15 to next available position in the heap. (i.e) 30 will be the child of 20. Do the Heapify Algorithm again to restore heap property  Z+RB &@ s *DpP 2 ' 6c ` ,$D 0 630 H  0޽h ? 3333RzR___PPT10ZR.!@׏+[DM' *= @B DQM' = @BA?%,( < +O%,( < +DL' (=%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%( DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(|D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(pD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(dD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(XD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(LD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(@D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4!D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%((#D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(%D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%('D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%()D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(,D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(.D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(0D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(2D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-6B'blinds(horizontal)*<3<*'D' =%(6DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-6B'blinds(horizontal)*<3<*'++0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+' ++0+' +<@ SK'(   # <8&  {CP r o c e d u r e F o r I n s e r t I o n I n t o H e a pD Df & C tA Pimg17\begin{Lalgorithm} {HeapInsert} {($H, n, \preceq, x$)} {A heap $(H,\preceq)$\ (r... ...if}\item \Lelsecuddle{$parent \gets 0$} \end{Lwhile}\end{Lgroup}\end{Lalgorithm}P ' < P FSource: http://planetmath.org/encyclopedia/HeapInsertionAlgorithm.htmlG G, H  0޽h ? 3333___PPT10i."Yy+D=' *= @B + mC 0(#&(    <,$ 0 \*I n p u t : new element 30 into the heap+ +2  6 `,$D  0 >    <Z6Z A,$  0 456 2  6P ,$D 0 >  2   6”P ,$D  0 >  2   6Ɣ  p,$D   0 >  2   6ʔ @p,$D  0 >  2   6ȔP 0 ,$D  0 >  2   6ДP,$D  0 >  2  6ԔP 0 ,$D  0 >  B @ s *D0@,$D   0B  s *D ,$D   0B @ s *D@P,$D  0B  s *DP ,$D   0B @ s *Dpp P,$D  0B  s *Dp pP,$D  0  <t۔t <    <`ޔg,$ 0 430   <j  Q,$ 0 440   <ٔ:!,$  0 411   <(g,$  0 425   < * ,$  0 430   <<jfsQ,$  0 415 2  68P P ,$D  0 >    <L  ,$  0 33   <j Q ,$  0 35 B @ s *DppP ,$D  0B   s *DpP ,$D  0XB !@ 0Dp @ x " 0` ` \  Heap is restored after inserting 30(the new element) into the heap. Time Complexity O (log n) We need to traverse in the heap till we find a position for the new element in the heap. (Position Found = Insert the element there RB #@ s *DpP 2 $ 6 ` ,$D 0 620 XB %@ 0D` XB & 0D@H  0޽h ? 3333RzR___PPT10ZR.# a+8oDM' )= @B DQM' = @BA?%,( < +O%,( < +DL' (=%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%( DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(|D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(pD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(dD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(XD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(LD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(@D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4!D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%((#D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(%D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%('D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%()D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(,D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(.D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(0D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(2D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$D' =%(6DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$++0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+$ ++0+$ +)hC SK# (      <pIz& a V H E A P E X T R A C T M A X! !f   <Mp `W,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ /2   6R `,$D  0 >     <QZ6Z A,$  0 456 2   6TP ,$D 0 >  2   6]P ,$D  0 >  2   6[  p,$D   0 >  2   6c @p,$D  0 >  2   6hP 0 ,$D  0 >  2   6DkP,$D  0 >  2   6oP 0 ,$D  0 >  B  @ s *D0@,$D   0B   s *D ,$D   0B  @ s *D@P,$D  0B   s *DP ,$D   0B  @ s *Dpp P,$D  0B   s *Dp pP,$D  0   <ut <     <xg,$ 0 425    <}j  Q,$ 0 440    <tt:!,$  0 411    <g,$  0 420    < * ,$  0 430    <\jfsQ,$  0 415 2   6P P ,$D  0 >     <Ē  ,$  0 33    <j Q ,$  0 35 B  @ s *DppP ,$D  0B   s *DpP ,$D  0 !  0ؚ @_2 Extracting the maximum element Maximum element is always at the root. In this example it is 56. Extract 56 and restore the heap property.  H   0޽h ? 3333~OvO___PPT10VO.$K+GlDJ' *= @B DJ' = @BA?%,( < +O%,( < +DI' (=%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%( D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%( DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(|D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(pD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(dD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(XD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(LD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(@D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(4!D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%((#D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(%D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%('D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%()D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(,D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(.D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(0D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(2D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(4D<' =A@BBBB0B%(D' =-m6Bbox(in)*<3<* D' =1:Bhidden*o3>+B#style.visibility<* %(+`+0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  ++0+  +]C E=$(    <ܘ } V H E A P E X T R A C T M A X! !f2  6P `,$D  0 >  2  6P ,$D 0 >  2   6P ,$D  0 >  2   6D  p,$D   0 >  2   6 @p,$D  0 >  2   6tP 0 ,$D  0 >  2  6P 0 ,$D  0 >  B @ s *D0@,$D  0B  s *D ,$D  0B @ s *D@P,$D  0B  s *DP ,$D   0B @ s *Dpp P,$D  0  <Dt <    <\g,$ 0 425   <j  Q,$ 0 430   < :!,$   0 411   <g,$   0 420   <@  ,$  0 415 2  6P P ,$D  0 >    <$  ,$  0 33   <4j Q ,$  0 35 B @ s *DppP ,$D  0B   s *DpP ,$D  0A ! 0  @_ Restoring the heap after extracting the maximum element. 40 is moved to the root 30 is moved in the place of 40. Running Time is O (lg n)   " <%$fx ,$ 0 440 B # 0D ,$D 0B $@ s *D@ p ,$D 0H  0޽h ? 3333FF___PPT10F.%p0E+ {`DC' *= @B DB' = @BA?%,( < +O%,( < +DB' (=%(%(D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%( DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(|D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(pD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(dD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(XD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(LD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(@D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4!D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%((#D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(%D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%('D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%()D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-6B'blinds(horizontal)*<3<*#D' =%(,DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-6B'blinds(horizontal)*<3<*$D' =%(.DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-6B'blinds(horizontal)*<3<*"++0+ ++0+ ++0+  ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+" +C %%19N%(    <^`B QH E A P D E L E T I O N f2  6c`,$D  0 >  2  6g  p,$D 0 >  2  6tjP0,$D  0 >  2  6n`P,$D   0 >  2   6xs,$D  0 >  2   6w0 p ,$D  0 >  2   6hz ,$D  0 >  B  @ s *D0,$D  0B   s *D0P,$D  0B @ s *D@@`,$D  0B  s *Dp ,$D   0B @ s *DP0 ,$D  0  <_p <    <lP7,$ 0 425   0`g,$ 0 430   <w,$   0 411   <,$   0 420   <H` -G ,$  0 415 2  6xp ,$D  0 >    <d0  ,$  0 33   0 l ,$  0 35 B @ s *D,$D  0B  s *Dp,$D  0  0   GDeleting the node 11.   <,$ 0 440 2 ! 6৚pJ,$D  0 >  2 " 6 j ,$D 0 >  2 # 6̮z,$D  0 >  2 $ 6< ,$D#  0 >  2 % 60Z @,$D&  0 >  2 ' 60 Z ,$D)  0 >  B (@ s *D zP,$D  0B ) s *DZ,$D   0B *@ s *D ,$D! 0B + s *D ,$D"  0 - <: <   . <Ú ,$ 0 425  / 0(ǚ*N,$ 0 430  0 <Țz F ,$$  0 35  1 <Ϛ  ,$%  0 420 2 3 6Ӛ0 : ,$D+  0 >   4 <֚ V w ,$*  0 33  5 0ښ` j G ,$,  0 415 B 6@ s *D J 0 ,$D'  0B 7 s *D: 0 ,$D(  0< 8 0ޚ X After Deleting the node 11. We move 5 to the top and move 15 to the right child of 5 Do the heapify algorithm to restore heap property ^$ 9 <pzW,$- 0 440 H  0޽h ? 3333~~___PPT10~.&v;n+uS:Dw' *= @B D>w' = @BA?%,( < +O%,( < +Duv' (=%(%(D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%( DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(|D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(pD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(dD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(XD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(LD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(@D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4!D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%((#D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(%D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%('D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%()D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(*DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(,DD' =A@BBBB0B%(D' =-u6Bdiamond(in)*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(4D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-m6Bbox(in)*<3<*!D' =%(6DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-6B'blinds(horizontal)*<3<*"D' =%(8DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-6B'blinds(horizontal)*<3<*.D' =%(:D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-m6Bbox(in)*<3<*#D' =%(<DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =-6B'blinds(horizontal)*<3<*/D' =%(>D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-m6Bbox(in)*<3<*(D' =%(t@D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-m6Bbox(in)*<3<*)D' =%(hBDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-6B'blinds(horizontal)*<3<**D' =%(\DD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =-m6Bbox(in)*<3<*+D' =%(PFD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-m6Bbox(in)*<3<*$D' =%(DHD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-m6Bbox(in)*<3<*0D' =%(8JD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-m6Bbox(in)*<3<*1D' =%(,LD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-m6Bbox(in)*<3<*%D' =%( ND1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-m6Bbox(in)*<3<*6D' =%(PD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =-m6Bbox(in)*<3<*7D' =%(RD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-m6Bbox(in)*<3<*'D' =%(SD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-m6Bbox(in)*<3<*4D' =%(UD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =-m6Bbox(in)*<3<*3D' =%(WD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =-m6Bbox(in)*<3<*5D' =%(YDT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*9%(D' =-6B'blinds(horizontal)*<3<*9D' =%([DD' =A@BBBB0B%(D' =-u6Bdiamond(in)*<3<*0D' =1:Bhidden*o3>+B#style.visibility<*0%(++0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+! ++0+" ++0+# ++0+$ ++0+% ++0+' ++0+. ++0+/ ++0+0 ++0+0 ++0+1 ++0+3 ++0+4 ++0+5 ++0+9 +WC +#(    <H& ^&H E A P D E L E T I O N (C o n t d)' 'f2  6hN `,$D  0 >  2  6,RP ,$D 0 >  2  6TP ,$D  0 >  2  6\Y  p,$D   0 >  2   6^ @p,$D  0 >  2   6aP 0 ,$D  0 >  2   6dP 0 ,$D  0 >  B  @ s *D0@,$D  0B   s *D ,$D  0B @ s *D@P,$D  0B  s *DP ,$D   0B @ s *Dpp P,$D  0  <jt <    <xmg,$ 0 425   <Tqj  Q,$ 0 430   <u:!,$   0 415   <tg,$   0 420   <|  ,$  0 415 2  6P P ,$D  0 >    <,  ,$  0 35   <hj Q ,$  0 33 B @ s *DppP ,$D  0B  s *DpP ,$D  0A  0 @P2 Heap is restored after Deleting the node 11 from the heap. Deletion takes O(log n) time to delete the node And call the heapify algorithm |   <P$fx ,$ 0 440 H  0޽h ? 3333AA___PPT10oA.'k+qjD=' *= @B D~=' = @BA?%,( < +O%,( < +D<' (=%(%(D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%( DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(|D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(pD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(dD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(XD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(LD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(@D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(4!D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%((#D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(%D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%('D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%()D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(*DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*++0+ ++0+ ++0+ ++0+ ++0+  ++0+  ++0+  ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ +hC $Hx(  H  H <Ɲ A,$ 0 j*Min Heap Heap Constructed from the array$+ f# H < ȝ,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ / H <tН,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ /2 H 6ԝ `,$D 0 611 2  H 6؝P ,$D 0 >  2  H 6۝P ,$D 0 >  2  H 6  p,$D 0 >  2  H 6L @p,$D  0 >  2  H 6P 0 ,$D  0 >  2 H 6|P,$D  0 >  2 H 6 P 0 ,$D  0 >  B H@ s *D0@,$D  0B H s *D ,$D 0B H@ s *D@P,$D 0B H s *DP ,$D 0B H@ s *Dpp P,$D 0B H s *Dp pP,$D 0 H <g,$ 0 420  H <j  Q,$ 0 415  H <:!,$ 0 430  H <`g,$ 0 425  H <  * ,$ 0 440  H < jfQ,$ 0 456 2 H 6tP P ,$D 0 >   H <  ,$ 0 33  H <dj Q ,$ 0 35 B  H@ s *DppP ,$D 0B !H s *DpP ,$D 0H H 0޽h ? 3333zQrQ___PPT10RQ.'pI++mGDL' *= @B DL' = @BA?%,( < +O%,( < +DK' (=%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =-6B'blinds(horizontal)*<3<* HD' =%( DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =-6B'blinds(horizontal)*<3<* HD' =%( DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =-6B'blinds(horizontal)*<3<* HD' =%( DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =-6B'blinds(horizontal)*<3<* HD' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =-6B'blinds(horizontal)*<3<* HD' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%(|DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%(pDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%(dDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%(XDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%(LDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%(@DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%(4!DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%((#DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%(%DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%('DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%()DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%(*DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%(,DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%(.DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%(0DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HD' =%(2DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =-6B'blinds(horizontal)*<3<* HD' =%(4DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*!H%(D' =-6B'blinds(horizontal)*<3<*!H+`+0+H ++0+H ++0+H ++0+H ++0+ H ++0+ H ++0+ H ++0+ H ++0+ H ++0+H ++0+H ++0+H ++0+H ++0+H ++0+H ++0+H ++0+H ++0+H ++0+H ++0+H +*aC  !L(  L L <$U  HMin Heap Comparing 30,5 and 3. 3 is less than 30 and 5. Swap 30 with 3 $I  f? L <W,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ / L <_,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ /2 L 6c `,$D  0 611 2 L 6gP ,$D 0 >  2  L 6jP ,$D  0 >  2  L 6p  p,$D   0 >  2  L 6to @p,$D  0 >  2  L 6vP 0 ,$D  0 >  2  L 6zP,$D  0 >  2 L 6D~P 0 ,$D  0 >  B L@ s *D0@,$D  0B L s *D ,$D   0B L@ s *D@P,$D  0B L s *DP ,$D   0B L@ s *Dpp P,$D  0B L s *Dp pP,$D  0 L < $ 420  L <j  Q,$ 0 415  L <Ќ:!,$   0 33  L < g,$  0 425  L < * ,$  0 440  L <jfQ,$  0 456 2 L 6P P ,$D  0 >   L <h * ,$  0 430  L <Tj Q ,$  0 35 B L@ s *DppP ,$D  0B L s *DpP ,$D  0XB !L@ 0D@@ H L 0޽h ? 3333II___PPT10I.'(H+6JDE' *= @B DKE' = @BA?%,( < +O%,( < +DD' (=%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-6B'blinds(horizontal)*<3<*LD' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-6B'blinds(horizontal)*<3<*LD' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-m6Bbox(in)*<3<*LD' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-6B'blinds(horizontal)*<3<*LD' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* L%(D' =-m6Bbox(in)*<3<* LD' =%( DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-6B'blinds(horizontal)*<3<*LD' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-m6Bbox(in)*<3<*LD' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-m6Bbox(in)*<3<*LD' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-6B'blinds(horizontal)*<3<*LD' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-m6Bbox(in)*<3<*LD' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* L%(D' =-m6Bbox(in)*<3<* LD' =%(|D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-m6Bbox(in)*<3<*LD' =%(pD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-m6Bbox(in)*<3<*LD' =%(dD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* L%(D' =-m6Bbox(in)*<3<* LD' =%(XD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-m6Bbox(in)*<3<*LD' =%(LD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-m6Bbox(in)*<3<*LD' =%(@D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* L%(D' =-m6Bbox(in)*<3<* LD' =%(4!D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-m6Bbox(in)*<3<*LD' =%((#D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* L%(D' =-m6Bbox(in)*<3<* LD' =%(%D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-m6Bbox(in)*<3<*LD' =%('D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-m6Bbox(in)*<3<*LD' =%()D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-m6Bbox(in)*<3<*LD' =%(*D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-m6Bbox(in)*<3<*LD' =%(,D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-m6Bbox(in)*<3<*LD' =%(.D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-m6Bbox(in)*<3<*LD' =%(0D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-m6Bbox(in)*<3<*L++0+L ++0+L ++0+L ++0+L ++0+ L ++0+ L ++0+ L ++0+ L ++0+ L ++0+L ++0+L ++0+L ++0+L ++0+L ++0+L ++0+L ++0+L ++0+L +,aC 0!P(  P P <ߟ X JMin Heap Comparing 3,20 and 25. 3 is less than 20 and 25. Swap 20 with 3.$K fC P <,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ / P <,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ /2 P 6 `,$D  0 611 2 P 68P ,$D 0 >  2  P 6P ,$D  0 >  2  P 6P  p,$D   0 >  2  P 6 @p,$D  0 >  2  P 6P 0 ,$D  0 >  2  P 6xP,$D  0 >  2 P 6H P 0 ,$D  0 >  B P@ s *D0@,$D  0B P s *D ,$D   0B P@ s *D@P,$D  0B P s *DP ,$D   0B P@ s *Dpp P,$D  0B P s *Dp pP,$D  0 P < 33  P <j  Q,$ 0 415  P <:!,$   0 420  P <g,$  0 425  P < * ,$  0 440  P <"jfQ,$  0 456 2 P 6!P P ,$D  0 >   P <) * ,$  0 430  P <d%j Q ,$  0 35 B P@ s *DppP ,$D  0B P s *DpP ,$D  0XB !P@ 0D@@H P 0޽h ? 3333II___PPT10I.' 0+6JDE' *= @B DKE' = @BA?%,( < +O%,( < +DD' (=%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-6B'blinds(horizontal)*<3<*PD' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-6B'blinds(horizontal)*<3<*PD' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-m6Bbox(in)*<3<*PD' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-6B'blinds(horizontal)*<3<*PD' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* P%(D' =-m6Bbox(in)*<3<* PD' =%( DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-6B'blinds(horizontal)*<3<*PD' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-m6Bbox(in)*<3<*PD' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-m6Bbox(in)*<3<*PD' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-6B'blinds(horizontal)*<3<*PD' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-m6Bbox(in)*<3<*PD' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* P%(D' =-m6Bbox(in)*<3<* PD' =%(|D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-m6Bbox(in)*<3<*PD' =%(pD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-m6Bbox(in)*<3<*PD' =%(dD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* P%(D' =-m6Bbox(in)*<3<* PD' =%(XD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-m6Bbox(in)*<3<*PD' =%(LD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-m6Bbox(in)*<3<*PD' =%(@D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* P%(D' =-m6Bbox(in)*<3<* PD' =%(4!D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-m6Bbox(in)*<3<*PD' =%((#D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* P%(D' =-m6Bbox(in)*<3<* PD' =%(%D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-m6Bbox(in)*<3<*PD' =%('D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-m6Bbox(in)*<3<*PD' =%()D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-m6Bbox(in)*<3<*PD' =%(*D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-m6Bbox(in)*<3<*PD' =%(,D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-m6Bbox(in)*<3<*PD' =%(.D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-m6Bbox(in)*<3<*PD' =%(0D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-m6Bbox(in)*<3<*P++0+P ++0+P ++0+P ++0+P ++0+ P ++0+ P ++0+ P ++0+ P ++0+ P ++0+P ++0+P ++0+P ++0+P ++0+P ++0+P ++0+P ++0+P ++0+P +aC @?T(  T "T <j  |JMin Heap Comparing 3,11 and 15. 3 is less than 11 and 15. Swap 3 with 11.K K #T <n,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ / $T <ls,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ /2 %T 6Pw `,$D  0 53 2 &T 6<{P ,$D 0 >  2 'T 6h~P ,$D  0 >  2 (T 6P  p,$D   0 >  2 )T 6 @p,$D  0 >  2 *T 6pP 0 ,$D  0 >  2 +T 6P,$D  0 >  2 ,T 6P 0 ,$D  0 >  B -T@ s *D0@,$D  0B .T s *D ,$D   0B /T@ s *D@P,$D  0B 0T s *DP ,$D   0B 1T@ s *Dpp P,$D  0B 2T s *Dp pP,$D  0 3T <8 411  4T <j  Q,$ 0 415  5T <螡:!,$   0 420  6T <$g,$  0 425  7T < * ,$  0 440  8T <ĪjfQ,$  0 456 2 9T 6ХP P ,$D  0 >   :T < * ,$  0 430  ;T <lj Q ,$  0 35 B +B#style.visibility<*#T%(D' =-6B'blinds(horizontal)*<3<*#TD' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*$T%(D' =-6B'blinds(horizontal)*<3<*$TD' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%T%(D' =-m6Bbox(in)*<3<*%TD' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*&T%(D' =-6B'blinds(horizontal)*<3<*&TD' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'T%(D' =-m6Bbox(in)*<3<*'TD' =%( DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*4T%(D' =-6B'blinds(horizontal)*<3<*4TD' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-T%(D' =-m6Bbox(in)*<3<*-TD' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.T%(D' =-m6Bbox(in)*<3<*.TD' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*/T%(D' =-6B'blinds(horizontal)*<3<*/TD' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0T%(D' =-m6Bbox(in)*<3<*0TD' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(T%(D' =-m6Bbox(in)*<3<*(TD' =%(|D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5T%(D' =-m6Bbox(in)*<3<*5TD' =%(pD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6T%(D' =-m6Bbox(in)*<3<*6TD' =%(dD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*)T%(D' =-m6Bbox(in)*<3<*)TD' =%(XD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1T%(D' =-m6Bbox(in)*<3<*1TD' =%(LD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*2T%(D' =-m6Bbox(in)*<3<*2TD' =%(@D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<**T%(D' =-m6Bbox(in)*<3<**TD' =%(4!D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*7T%(D' =-m6Bbox(in)*<3<*7TD' =%((#D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+T%(D' =-m6Bbox(in)*<3<*+TD' =%(%D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8T%(D' =-m6Bbox(in)*<3<*8TD' =%('D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+B#style.visibility<*=T%(D' =-m6Bbox(in)*<3<*=TD' =%(*D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*,T%(D' =-m6Bbox(in)*<3<*,TD' =%(,D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*:T%(D' =-m6Bbox(in)*<3<*:TD' =%(.D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*9T%(D' =-m6Bbox(in)*<3<*9TD' =%(0D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;T%(D' =-m6Bbox(in)*<3<*;T++0+#T ++0+$T ++0+%T ++0+&T ++0+'T ++0+(T ++0+)T ++0+*T ++0++T ++0+,T ++0+4T ++0+5T ++0+6T ++0+7T ++0+8T ++0+9T ++0+:T ++0+;T +aC P!X(  X X <8 X |JMin Heap Comparing 5,20 and 30. 5 is less than 20 and 30. Swap 5 with 20.K K X <,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ / X <D,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ /2 X 6 `,$D  0 53 2 X 6P ,$D 0 >  2  X 6 P ,$D  0 >  2  X 6  p,$D   0 >  2  X 6 @p,$D  0 >  2  X 6P 0 ,$D  0 >  2  X 6P,$D  0 >  2 X 6P 0 ,$D  0 >  B X@ s *D0@,$D  0B X s *D ,$D   0B X@ s *D@P,$D  0B X s *DP ,$D   0B X@ s *Dpp P,$D  0B X s *Dp pP,$D  0 X <$ 411  X <'j  Q,$ 0 415  X <+:!,$   0 35  X <L.g,$  0 425  X <`3 * ,$  0 440  X <7jfQ,$  0 456 2 X 6;P P ,$D  0 >   X <> * ,$  0 430  X <DBj Q ,$  0 420 B X@ s *DppP ,$D  0B X s *DpP ,$D  0XB !X 0DpP H X 0޽h ? 3333II___PPT10I.'.+6JDE' *= @B DKE' = @BA?%,( < +O%,( < +DD' (=%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-6B'blinds(horizontal)*<3<*XD' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-6B'blinds(horizontal)*<3<*XD' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-m6Bbox(in)*<3<*XD' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-6B'blinds(horizontal)*<3<*XD' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* X%(D' =-m6Bbox(in)*<3<* XD' =%( DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-6B'blinds(horizontal)*<3<*XD' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-m6Bbox(in)*<3<*XD' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-m6Bbox(in)*<3<*XD' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-6B'blinds(horizontal)*<3<*XD' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-m6Bbox(in)*<3<*XD' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* X%(D' =-m6Bbox(in)*<3<* XD' =%(|D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-m6Bbox(in)*<3<*XD' =%(pD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-m6Bbox(in)*<3<*XD' =%(dD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* X%(D' =-m6Bbox(in)*<3<* XD' =%(XD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-m6Bbox(in)*<3<*XD' =%(LD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-m6Bbox(in)*<3<*XD' =%(@D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* X%(D' =-m6Bbox(in)*<3<* XD' =%(4!D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-m6Bbox(in)*<3<*XD' =%((#D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* X%(D' =-m6Bbox(in)*<3<* XD' =%(%D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-m6Bbox(in)*<3<*XD' =%('D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-m6Bbox(in)*<3<*XD' =%()D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-m6Bbox(in)*<3<*XD' =%(*D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-m6Bbox(in)*<3<*XD' =%(,D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-m6Bbox(in)*<3<*XD' =%(.D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-m6Bbox(in)*<3<*XD' =%(0D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-m6Bbox(in)*<3<*X++0+X ++0+X ++0+X ++0+X ++0+ X ++0+ X ++0+ X ++0+ X ++0+ X ++0+X ++0+X ++0+X ++0+X ++0+X ++0+X ++0+X ++0+X ++0+X +6aC  ` \(  \ \ <{ SH bMin Heap Comparing 5,11 and 25. 5 is less than 11 and 25. Swap 5 with 11. Heap Property Restoredc c \ <,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ / \ <,$ 0 `.I n p u t : 11, 20, 15, 30, 25, 40, 56, 3, 5/ /2 \ 6 `,$D  0 53 2 \ 6P ,$D 0 >  2  \ 6XP ,$D  0 >  2  \ 6  p,$D   0 >  2  \ 6 @p,$D  0 >  2  \ 6PP 0 ,$D  0 >  2  \ 6蠥P,$D  0 >  2 \ 6P 0 ,$D  0 >  B \@ s *D0@,$D  0B \ s *D ,$D   0B \@ s *D@P,$D  0B \ s *DP ,$D   0B \@ s *Dpp P,$D  0B \ s *Dp pP,$D  0 \ < 35  \ <謥j  Q,$ 0 415  \ <:!,$   0 411  \ <g,$  0 425  \ <踥 * ,$  0 440  \ <jfQ,$  0 456 2 \ 6P P ,$D  0 >   \ <Xå * ,$  0 430  \ <Dǥj Q ,$  0 420 B \@ s *DppP ,$D  0B \ s *DpP ,$D  0XB  \@ 0DH \ 0޽h ? 3333II___PPT10I.'8`+6JDE' *= @B DKE' = @BA?%,( < +O%,( < +DD' (=%(%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-6B'blinds(horizontal)*<3<*\D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-6B'blinds(horizontal)*<3<*\D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-m6Bbox(in)*<3<*\D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-6B'blinds(horizontal)*<3<*\D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* \%(D' =-m6Bbox(in)*<3<* \D' =%( DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-6B'blinds(horizontal)*<3<*\D' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-m6Bbox(in)*<3<*\D' =%( D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-m6Bbox(in)*<3<*\D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-6B'blinds(horizontal)*<3<*\D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-m6Bbox(in)*<3<*\D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* \%(D' =-m6Bbox(in)*<3<* \D' =%(|D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-m6Bbox(in)*<3<*\D' =%(pD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-m6Bbox(in)*<3<*\D' =%(dD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* \%(D' =-m6Bbox(in)*<3<* \D' =%(XD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-m6Bbox(in)*<3<*\D' =%(LD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-m6Bbox(in)*<3<*\D' =%(@D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* \%(D' =-m6Bbox(in)*<3<* \D' =%(4!D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-m6Bbox(in)*<3<*\D' =%((#D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* \%(D' =-m6Bbox(in)*<3<* \D' =%(%D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-m6Bbox(in)*<3<*\D' =%('D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-m6Bbox(in)*<3<*\D' =%()D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-m6Bbox(in)*<3<*\D' =%(*D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-m6Bbox(in)*<3<*\D' =%(,D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-m6Bbox(in)*<3<*\D' =%(.D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-m6Bbox(in)*<3<*\D' =%(0D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-m6Bbox(in)*<3<*\++0+\ ++0+\ ++0+\ ++0+\ ++0+ \ ++0+ \ ++0+ \ ++0+ \ ++0+ \ ++0+\ ++0+\ ++0+\ ++0+\ ++0+\ ++0+\ ++0+\ ++0+\ ++0+\ +V@ mep`(  ` ` <D T pBuild  Max Heap O(n) Max  Heapify O(lg n) Heap Sort O(n lg n) Heap  Maximum O(1) Heap  Extract Max O(lg n) Heap- Increase Key O(lg n) Max-Heap-Insert O( lg n) Heap Search O(n) Heap Delete O(lg n)   0( ` B &  o5H E A P S O R T ( T I M E C O M P L E X I T Y)6 6fH ` 0޽h ? 3333___PPT10i.'`:j+D=' *= @B +3"@ J!B!=@ (  @x @ <A p?? p p @ <0 m#H E A P S O R T A N A L Y S I S.$ f"fA @ =@ #">2 @N 2@ B?VNF___PPT9(  v33.21928< (" dcC @`Q 1@ B?@VNF___PPT9(  y 10000000000< (" d cC @`N 0@ Bh5? VNF___PPT9(  v29.89735< (" dcC @`P /@ B8?@ VNF___PPT9(  x 1000000000< (" d cC @`N .@ B7?  VNF___PPT9(  v26.57542< (" dcC @`O -@ B$T?@  VNF___PPT9(  w 100000000< (" d cC @`M ,@ BlU?  VNF___PPT9(  u23.2535<(" dcC @`N +@ BTg?@  VNF___PPT9(  v10000000< (" dcC @`N *@ B`j?4  VNF___PPT9(  v19.93157< (" dcC @`M )@ B{?@4  VNF___PPT9(  u1000000<(" dcC @`N (@ B?N 4 VNF___PPT9(  v16.60964< (" dcC @`L '@ B$K?@N 4 VNF___PPT9(  t100000<(" dcC @`N &@ B?h N VNF___PPT9(  v13.28771< (" dcC @`K %@ B?@h N VNF___PPT9(  s10000<(" dcC @`N $@ Bp?h VNF___PPT9(  v9.965784< (" dcC @`J #@ Bl?@h VNF___PPT9(  r1000<(" dcC @`N "@ B?VNF___PPT9(  v6.643856< (" dcC @`I !@ Bʧ?@VNF___PPT9(  q100<(" dcC @`N  @ B§?VNF___PPT9(  v3.321928< (" dcC @`H @ B$ߧ?@VNF___PPT9(  p10<(" dcC @`n @ B?VNF___PPT9(  n lg n<(" dcC @`G @ Bh?@VNF___PPT9(  on<(" dcC @`HB 3@ C @HB 4@ C @HB 5@ C @@HB 6@ C H @ 0޽h ? 3333___PPT10i.'0P>+D=' *= @B +@ /'(    6  G S T R E N G T H S All data types. Medium Complexity W E A K N E S S Not a stable sort since every insert or delete operation heapify must be done. fO U ,P   B `&STRENGTHS AND WEAKNESS OF HEAP SORT' 'fH  0޽h ? 3F433___PPT10i. L=+D=' *= @B +@ H(  v  0p H Worst case running time of Quick Sort is O(n2) while the Worst Case running time of Heap Sort is same as the best and average case O(n lg n). Heap Sort algorithm is used where there is unacceptable complexity for the running time of the algorithm Embedded systems with real-time constraints often uses Heap Sort algorithm  j  L C  <  <    <   v,H E A P S O R T V S Q U I C K S O R T.- f+fH  0޽h ? 3333___PPT10i.'@Ny+D=' *= @B + -@ 7,/,5+(  ) p  #"6* ___U_____p  B0?  O---  @`  Bl9?  ZAlready sorted  @`  B;?  ZAlready sorted  @`  BDL?  ZAlready sorted  @`  BhV?p  e When Best* (  @`  B$_?C   O---  @`  Ba? C   ZReverse Sorted  @`  B0r? C  ZReverse sorted  @`  Bl{?C  ZReverse sorted  @`  B?pC   f When Worst* (  @`  BȎ? C  _O (n log n) or O(1)  @`  B? C  PO(1)  @`  Bl? C  PO(1)  @`  BP? C  PO(1)  @`  B<?p C  g Best Copies* (  @`  B죩?i   W O (n log n)    @`  B8? i   s O(n2)or O (n)4    @`  Bѩ? i  kO(n2)4   @`  B˩?i  QO (n)  @`  B@?pi   i Best Compares*(  @`  B?i  W O (n log n)    @`  B@? i  kO(n2)4   @`  B?  i  QO (n)  @`  B<? i  kO(n2)4   @`  BP?pi  h Worst Copies* (  @`  B ? W O (n log n)    @`  B?  kO(n2)4   @`  B3?   kO(n2)4   @`  B<>?  kO(n2)4   @`  BH?p jWorst Compares*( @`  BTQ? W O (n log n)    @`  B[?  lO (n2)4   @`  Bd?   QO (n)  @`  Bn?  kO(n2)4   @`  Bh?p g Avg. Copies* (  @`  B? Z O (n log n)  (  @`  B?  nO(n2) 6(  @`  B?   mO(n2)6(  @`  B?  r e.g. O(n2)6 (  @`  B?p i Avg. Compares*(  @`  B? d Merge Sort(    @`  B?  e Bubble Sort(    @`  B4ɪ?   hSelection Sort(  @`  B`Ӫ?  hInsertion Sort(  @`  Bܪ?p T  @`HB  C pHB  C pHB  C ppHB  C   <ߪ0 c-R U N N I N G T I M E C O M P L E X I T Y..f  00v KSource: http://ccl.northwestern.edu/tisue/cs311/spring-01/sq2/answers.html L DU 0JH  0޽h ? 33___PPT10i.q Ê+D=' *= @B +@@ @(    <  JK E Y P O I N T S  fJ  <&~  Sorting in O (n log n) Most of the sorting algorithm will take at least O(n lg n) steps to sort n items. Reason: Comparison of the data items in the array Moving the data items from one position to another Working with extra space (Sometimes in Quick Sort Worst Case)  A    A3vMH  0޽h ? 333380___PPT10.c.@  (    <`Z  V WiV  08tZP Chttp://www.devx.com/vb2themax/Article/19900 http://www.fearme.com/misc/alg/node42.html http://www.informatik.uni-stuttgart.de/ipvr/sgs/lehre/seminare/famous_algorithms_ws03_04/01_abstract.html http://www.cs.princeton.edu/~rs/talks/Montreal.pdf http://www.cs.colorado.edu/~karl/2270.fall03/sorting.html http://www.cs.rit.edu/~jdb/cs2/qsortAndMSort.pdf http://www.cs.princeton.edu/courses/archive/spring04/cos226/lectures/radix.4up.pdf R.L. Wainwright, A Class of Sorting Algorithms based on Quicksort, Communications of the ACM, Vol. 28, No. 4, April 1985, pgs. 396-402. C.R. 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Output: INSIDE ( a Boolean variable, True if q is inside P, and false otherwise) Count 0; for all edges ei of the polygon do if the line x = x0 intersects ei then yi y coordinate of the intersection between lines ei and x=x0; if yi > y0 then Count Count +1; if count is odd then INSIDE TRUE; else INSIDE FALSE ! %            P       1                                 ` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> f(    6̙  P   T Click to edit Master title style! !  04ϙ     RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0ՙ  ``   >*  0ڙ  `    @*  0lߙ  `    @*H  0޽h ? 3f Default Design 0 zr0@ (  @ @ 0  P     P*   @ 0       R*  d @ c $ ?    @ 0   @   RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S @ 6L  `P    P*   @ 6\  `    R*  H @ 0޽h ? ̙3380___PPT10.0 V  p(  l  C  p0@     C Ϲ  p  ITOPICS Preliminaries Point in a Polygon Polygon Construction Convex Hulls@C q!CH  0޽h ? ̙33y___PPT10Y+D=' = @B +a  P ( $R  l  C      l  C \ p0   H  0޽h ? ̙33y___PPT10Y+D=' = @B +!  H@`(  l  C @'     ,  0) p A point p is represented as a pair of coordinates (x,y) A line is represented by a pair of points A path is a sequence of points p1,p2, . . . pn and the line segments connecting them, p1-p2, p2-p3, . . . , p k-1-pk. A closed path whose last point is the same as the first is a polygon. A simple polygon is one whose corresponding path does not intersect itself. It encloses a region in the plane. A Convex Polygon is a polygon such that any line segment connecting two points inside the polygon is itself entirely in the polygon. The convex hull of a set of points is defined as the smallest convex polygon enclosing all the given points.(3&    *        <bw ]H  0޽h ? ̙33y___PPT10Y+D=' = @B +  &p$A( !Ƚx   c $PM  0 yA line segment connecting two points: The points are inside the polygon The line segment is not entirely in the polygon zz:B 4 3 >Q#:B 5 3 >] jF2 6 S 333>93F2 7 S 333>P:B 9 3 >N:B : 3 >^ L    #  A TB  c $DjJ TB B c $DjJh N2  S 333jJD<N2   S 333jJV TB   c $DjJh  N2   S 333jJ  *N2   S 333jJ: r N2   S 333jJ t N2  S 333jJN2  S 333jJ~xP N2  S 333jJ,BTB  c $DjJ  TB  c $DjJ< ^TB  c $DjJH `TB  c $DjJ`zZTB B c $DjJdN2  S 333jJJ~&N2  S 333jJ8 ZB  s *DjJB  0(Q `   OThis is not a convex polygon  2pB  HDoPp ` F2 < S 333>P` F2 = S 333>) > 0V   \p(x1,y1)8 2 ? 0T\  @F \q(x2,y2)8 2 @ 0a   \r(x3,y3)8 2 A 0g    \s(x4,y4)8 2H  0޽h ? ̙33y___PPT10Y+D=' = @B +   7( hr@y@ l  C |n 0   ^L !"/ # ` TB  c $DjJ!"!TB B c $DjJz!"%TB  c $DjJJ 4%z%TB  c $DjJJ %)TB  B c $DjJ< ) +TB   c $DjJ< f*+TB   c $DjJ*"+TB   c $DjJ !&/TB   c $DjJ.V D/TB  c $DjJV 0& 6/TB  c $DjJ >& R/TB  c $DjJ ,~6/TB  c $DjJb,P/TB  c $DjJP+"/N2  S 333jJ#bb$N2  S 333jJn * F+N2  S 333jJ"))%  0r  0% Given a simple polygon P, and a point q, determine whether the point is inside or outside the polygon. (a non-convex polygon) * 2H  0޽h ? ̙33y___PPT10Y+D=' = @B +   80(   l  C } P       <T Z`,$ 0 |HThis does not work if the line passes through terminal points of edgesIIH  0޽h ? ̙33  ___PPT10` + D ' = @B D ' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*?%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*?D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*?D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*?I%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*?ID' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*?I+8+0+  +    S( <  R Z j!4  3  0TB  c $DjJ !b&$N2   S 333jJ #` $N2   S 333jJZ <), )N2   S 333jJ / f0N2   S 333jJ&*^+N2   S 333jJh%&N2   S 333jJ..N2   S 333jJ)b8*TB  B c $DjJ B$ .)TB   c $DjJ ) /TB  c $DjJ *0TB  c $DjJ)+TB   c $DjJ)d.TB  B c $DjJV% |.N2   S 333jJx! "TB   c $DjJ!%N2   S jJ|,N-ZB   s *DjJj!4q  0X p It takes constant time to perform an intersection between two line segments. The algorithm computes n such intersections, where n is the size on the polygon. Total running time of the algorithm, O(n). , 2H  0޽h ? ̙33y___PPT10Y+D=' = @B +P    $S (  $ $ 0  0 u Constructing a Simple Polygon Given a set of points in the plane, connect them in a simple closed path. Consider a large circle that contains all the points. Scan the area of C by a rotating line. Connect the points in the order they are encountered in the scan.PjKL FRN> $# 0<2  $ # jJpRN><2  $ # <<2  $ # <2  $ # z <2  $ # Rn<2 $ # HB<2 $ # 68<2 $ # z<2 $ # vt<2 $ # 333frNB $ S DFH$H(2 $  l 0 p  $0 p ,$D 0`B $ 0DP 0 `B $ 0D0 @ P ` `B $ 0D0 `   `B $ 0D@  0 `B $ 0D `B $ 0D P p `B $B 0D p `B $B 0D P H $ 0޽h ? ̙335-___PPT10 +b#D' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-o6Bbox(out)*<3<*$+a  ((   ( ( 0( N Procedure Simple_Polygon Input : p1,p2, . . . pn (points in the polygon) Output : P ( a simple polygon whose vertices p1,p2, . . . pn are in some order), 1. for i 2 to n 2. ai angle between line p1-pi and the x-axis; 3. sort the points according to the angles (use the corresponding priority for the point and do a heapsort) 4. P is the polygon defined by the list of points in the sorted order. Complexity : Complexity of the sorting algorithm.     '       H ( 0޽h ? ̙33y___PPT10Y+D=' = @B +<  c[,( 01 , , 0  [Convex Hulls The convex hull of a set of points is defined as the smallest convex polygon enclosing all the points in the set. The convex hull is the smallest region encompassing a set of points. A convex hull can contain as little as three and as many as all the points as vertices. Problem Statement : Compute the convex hull of n given points in the plane. There are two algorithms Gift Wrapping O(n2) Graham's Scan O(nlogn) + 2  w H , 0޽h ? ̙33y___PPT10Y+D=' = @B +)  me  -0@(  0: 0 3 P  0 c $ ` +  Zp1<f3f3f3T8  -0S N2 0 S f3jJ%  N2 0 S f3jJO g N2  0 S f3jJ0ON2  0 S f3jJ"  k N2  0 S f3jJ 6& N2 0 S f3jJ(   0 c $   9  Zp2<f3f3f3 0 c $ K V R Zp3<f3f3f3 0 c $X Z Zp4<f3f3f3 0 c $    Zp5<f3f3f3 0 c $ G  Zp6<f3f3f3N2 0 S f3jJ ( N2 0 S f3jJ p N2 0 S f3jJ  h N2 0 S f3jJs & N2 0 S f3jJ" L N2 0 S f3jJ Z N2 0 S f3jJ]Gl  @p '0@ p,$D 0lB 0 <Df3oP plB  0 <Df3o @rB !0 BDf3op  &0 0 P ML 2 B (0 s *Df3op@,$D 0B )0@ s *Df3op ,$D 0B *0 s *Df3Ԕ`@@ ,$D 0B +0 s *Df3ԔP PP ,$D 0B ,0@ s *Df3ԔP ` ,$D 0H 0 0޽h ? ̙33___PPT10+,D' = @B DK' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'0%(Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(0%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(0D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(0Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)0%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*)0D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*)0Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**0%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<**0D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<**0Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+0%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*+0D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+0Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,0%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,0D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,0++  RJ4(   4 4 0    G JProcedure Gift_Wrapping(p1,p2, . . . pn) Input : p1,p2, . . . pn ( a set of points in the plane) Output : P (the convex hull of p1,p2, . . . pn ) 1. P {0} or e; 2. p a point in the set with the largest x-coordinate; 3. Add p to P; 4. L line containing p and parallel to the x-axis; 5. while |P| < n do 6. q point such that the angle between the line -p-q- and L is minimal among all points; 7. add q to P; 8. L line -p-q-; 9. pq;  2          !     G2 xH 4 0޽h ? ̙33y___PPT10Y+D=' = @B +     @8  (   8 8 0B  f &Graham's Scan: Given a set of n points in the plane, ordered as in the algorithm Simple Polygon, we can find a convex path among the first k points whose corresponding convex polygon encloses the first k points.6 8 s *F oPe  Lqm-1,RB 8 s *Do0jIL2 8 c $omX L2  8 c $o0 # L2  8 c $of L2  8 c $o t L2  8 c $o T6RB  8 s *Do# X G RB 8 s *Dou { RB 8 s *Do RB 8@ s *DoF HL2 8 c $o 4|RB 8 s *Do57>  8 s *P o ; Hp1* 8 s *R o0 Jpk+1*L2 8 c $o,wL2 8 c $oJ } RB 8 s *Do# i XB 8 0Do `XB 8 0Do# DRB 8@ s *DoXR  8 s *Y oZ   Hqm*H 8 0޽h ? ̙33y___PPT10Y+D=' = @B + @1PD`"(  DL2 D c $333jJ_kL2 D c $333jJL2 D c $333jJ!PL2 D c $333jJ3qL2  D c $333jJsL2 D c $333jJ  D c $` d-  Np10 L2 D c $333jJ7L2 D c $333jJB D c $e - K  Np20  D c $j  ? Np30  D c $o q Np40  D c $u 0 Np50   D c $Lr  Np60  !D c $( 0 Np70  "D c $ e Np80 pB #D HDoEjB %D BDo &D 6 ! Gq1:p1 q2:p2 q3:p3 m:3 2pB )D HDo0 6D s *@ jJ  Np40 L2 DD c $333jJYL2 +D c $333jJ{ L2 ,D c $333jJ @ L2 -D c $333jJp 1 L2 .D c $333jJ  L2 /D c $333jJ  L2 0D c $333jJ-l 1D s * jJ =  Nq10 L2 3D c $333jJd   4D s * jJE=[  Bq2$  5D s * jJ 0O[ Nq30  7D s *̣ jJU +  Np50  8D s * jJ  [ Np60  9D s * jJU )  Np70  :D s * jJ  Np80 dB ;D <Do  jB >D BDo @ jB ?D BDo ^B AD 6D  ED 6X    E-q2-q3- and -q3-p4- 2 GD < jJp 6k =4 2L  ` MD# @P 2 JD  C|KENGJ|KQo `Tܟ%`T|K`Tܟ%`T|K|K%`T|K|K`2 KD CoyENH Qo `TKoy`TKoy`TKoy`T l2 LD <og` ND 6  p &  ^The angle is < 180 2  OD 6  6 B-q (m-1)-qm- and -q m-pk - 180o8" 2 H D 0޽h ??0JDKDLD 3fy___PPT10Y+D=' = @B +& 7IH (  H H c $ @_ Nq40 F2 H S 333jJF2 H S 333jJDF2 H S 333jJ'p0F2 H S 333jJ`hF2 H S 333jJ+ t F2  H S 333jJ=F2  H S 333jJJ\  H c $   Nq10 F2  H S 333jJTf  H c $p   4q2 H c $ @{ Nq30  H c $ E Np50  H c $L PoK Np60  H c $ E Np70  H c $ "A  Np80 dB H <Do_H jB H BDop dB H <Do0` jB H@ BDo ` H 0 i ] qm:p4 m:48 2  H 0  I-q3-q4- and -q4-p5- 2 H c $l  ?K  Nq40 F2 H S 333jJPF2 H S 333jJ  >F2 H S 333jJA 0 F2  H S 333jJ (q F2 !H S 333jJ4 F2 "H S 333jJ FR F2 #H S 333jJm S $H c $  ^}?  Nq10 F2 %H S 333jJ &  &H c $ } 4q2 'H c $ `p Nq30  (H c $   Mk  Nq50  )H c $ ` /  Np60  *H c $"  J i  Np70  +H c $((  @ Np80 dB ,H <DoU dB .H <Do@  jB /H@ BDo` jB 0H BDo0p P  1H 0`.  p,$ 0 ] qm:p5 m:58 2 ^B 3H 6Do@ 5H 0P4  `  4k =5 2 6H 08    3k=6 2 7H # B%C DEFXf3:: nJ/nu`S?CEiL7  $ 9 G  l z  e  3 %        M4 @ LS c fb $ , z yD u4.)%),@                     2 +B#style.visibility<*1H%(D' =-o6Bbox(out)*<3<*1HD' =%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*1H %(D' =-o6Bbox(out)*<3<*1H +8+0+1H  +$7 RJ5DL R(  L# L 0] ` ,$ 0 YAngle between -q4-q5- and -q5-p6- is greater than 180 Therefore m = m-1 = 4 We skip p5Z 2Z, =L 0Dc  ` D,$ 0 ZAngle between -q3-q4- and -q4-p6- is greater than 180 Therefore m = m-1 = 3 We skip p4 "[ 2Zj ?L # BBCmDEF<f3%%>U}S<d,^ m>1":Nfz  ` u  M 2    ' N9 R d B r #u x &= 5 DMF&&@             v  L c $h  Nq40 F2 L S 333jJp0yF2 L S 333jJF2 L S 333jJF2 L S 333jJ@QF2 L S 333jJc F2  L S 333jJ&F2  L S 333jJ3<  L c $o >]  Nq10 F2  L S 333jJ4r  L c $(u ] { P 4q2 L c $Xy Po+ Nq30  L c $\~ %-K` Nq50  L c $܂ + Np60  L c $D %*I` Np70  L c $`  Np80 dB L <DoudB L <Do  jB L@ BDo@jB L BDoPdB L <Do` j2 L Bo`2 L CoyENH Qo `TKoy`TKoy`TKoy`Tqd2 L <op @L 6   3k=6 2 :L 6 @ ` I-q3-q4- and -q4-p6- 2F2 $L S 333jJ/ TnF2 %L S 333jJq 7 F2 &L S 333jJ x F2 'L S 333jJ ;A F2 (L S 333jJC M  *L c $` 4 o  Nq10  ,L c $Ȣ  4q2 -L c $,  Nq30  /L c $ `   Np60  0L c $D   Np70  1L c $ 5 2Qp Np80 dB 2L <Do oX dB 3L <Dop0 jB 5L BDo p8 P  DLP  "L c $ @@_{  Nq40 N2 #L S 333jJON2 )L S 333jJZr2 7L BoP 2 8L CoyENH Qo `TKoy`TKoy`TKoy`T  l2 9L <oP jB ;L BDo AL 6¹  ~  3k=6 2H L 0޽h ?o`LLL 7L 8L 9L 3fzr___PPT10R+|D' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-o6Bbox(out)*<3<*LD' =%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*L7%(D' =-o6Bbox(out)*<3<*L7D' =%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*L7O%(D' =-o6Bbox(out)*<3<*L7OD' =%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*LOZ%(D' =-o6Bbox(out)*<3<*LOZD' =%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*=L%(D' =-o6Bbox(out)*<3<*=LD' =%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*=L7%(D' =-o6Bbox(out)*<3<*=L7D' =%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*=L7O%(D' =-o6Bbox(out)*<3<*=L7OD' =%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*=LOZ%(D' =-o6Bbox(out)*<3<*=LOZ+p+0+L  ++0+=L  +   P j ( u PF2 P S 333jJ?D~F2 P S 333jJ'pF2 P S 333jJh0 F2 P S 333jJ+ t QF2  P S 333jJS= F2  P S 333jJJ   P c $D D   Nq10   P c $$J   4q2  P c $M   Nq30  P c $R Po Np60  P c $\X  Np70  P c $x] E "A  Np80 dB P <Do%_ H jB P BDo` jB P BDopp 02 P CoyENH Qo `TKoy`TKoy`TKoy`T` wd2 P < p/ P <c  T,$ 0 YAngle between -q2-q3- and -q3-p6- is less than 180 Therefore m = m+1 = 4 and q4 = p6 Z 2Z P 0h @ 3k=6 2H P 0޽h ?/ PP 3fb Z ___PPT10: +oD ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-o6Bbox(out)*<3<*PD' =%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*P5%(D' =-o6Bbox(out)*<3<*P5D' =%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*P5M%(D' =-o6Bbox(out)*<3<*P5MD' =%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*PMY%(D' =-o6Bbox(out)*<3<*PMY+8+0+P  + %-T@|(  TL2 T c $333oOL2 T c $333oL2 T c $333oQ@L2 T c $333o# aL2 T c $333oc&L2 T c $333o3  T s *t oT>]  Nq10   T s *y o] {  4q2  T s *} oPo Nq30   T s * o Nq40   T s * o*I  Np70  T s *, oU  Np80 dB T <Do5jB T BDo@0 `p2 T Ho `dB T <Do  jB T BDo0 T 0`  p  3k=7 2F2 T S 333jJ $F2 T S 333jJ P F2 T S 333jJ@ H F2 T S 333jJc T F2 T S 333jJ f F2 T S 333jJ*s< T c $  ~  Nq10   T c $@ P  4q2 !T c $ + Nq30  "T c $ 0 O + Nq40  #T c $ȫ % j `  Nq50  $T c $  ! Np80 dB %T <Dou ?( dB (T <Do@`` jB )T BDo@@p dB *T <Do`jB +T BDo0 P P ,T 6  3k=8 2H T 0޽h ?T 3fy___PPT10Y+D=' = @B +    X (   XL2 X c $333oZDL2 X c $333o'pL2 X c $333o PH L2 X c $333o|$ L2 X c $333oF 8 L2  X c $333or #  X c $x 5 Z  Nq10   X c $  e 4q2  X c $   Nq30   X c $ | Nq40  X c $ ^3 Nq50  X c $$ a |  Nq60 dB X <Do N dB X <Do( , xdB X <Do dB X <Do8dB X <Do8r dB X <Doh r  H X 0޽h ? 3fy___PPT10Y+D=' = @B +  + # < ( ,w < < 0   V Procedure Graham's Scan(p1,p2, . . . pn) Input : p1,p2, . . . pn(a set of points in the plane) Output : q1,q2, . . . qn( the convex hull of p1,p2, . . . pn) p1 the point in the set with the largest x-coordinate (and smallest y-coordinate if there are more than one point with the same x-coordinate) Construct Simple Polygon and arrange points in order Let order be p1,p2, . . . pn q1 p1; q2 p2; q3 p3; (initially P consists of p1,p2,and p3) m 3; for k 4 to n do while the angle between lines -q m-1-qm- and -q m-pk - 180o do m m-1; m m+1; qm pk; r8X,                 4XD                      pHH < < z   Y[Internal to the polygon]$H < 0޽h ? ̙33y___PPT10Y+D=' = @B +rdP$v') 1AJW[aow`{4h13K^ ? v Oh+'0X `h  Computational Geometry Mohan Kumar Mohan Kumar11aMicrosoft PowerPointy@!@ La@0GH g  (  3-- @ !--'@Times New Roman-. 2 CSE5311 Kumar  ."System-@Times New Roman-.  2 1 .-@Times New Roman-. (2 Computational Geometry,3!""02.--- @ !,<[---- $[<[hh<--'@Arial-. 2 bfTOPICS .--fff-@Monotype Sorts-.  2 fq .-@Arial-. 2 p Preliminaries   .-@Monotype Sorts-.  2 fq .-@Arial-. "2 pPoint in a Polygon     .-@Monotype Sorts-.  2 fq .-@Arial-. %2 pPolygon Construction    .-@Monotype Sorts-.  2 fq .-@Arial-. 2 p Convex Hulls  .-՜.+,0     On-screen Show Curtin University`wA Times New RomanArialMonotype SortsSymbolDefault DesignComputational GeometryGeometric Algorithms Preliminaries:Slide 40Determining whether a point is inside a polygonSlide 6Slide 7Slide 8Slide 9 Slide 10 Slide 11 Slide 12 Slide 13 Slide 14 Slide 15 Slide 16 Slide 17 Slide 18 Slide 19 Slide 20  Fonts UsedDesign Template Slide TitlesRoot EntrydO)^Current UserJSummaryInformation(PowerPoint Document(`w&_?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root EntrydO)Current UserSummaryInformation(PowerPoint Document(`wDocumentSummaryInformation8cse5311-old/cg_string_match_ex.doc010064400017600000145000000530001004101104400200740ustar00kumarfaculty00000400000024ࡱ> &(%5@ 0 bjbj22 "XX:::::::N N .C R ~:::   ::   @:,::x Yf#j@ 0 J.  xNN:::: :x$ NN NNCSE 5311 Spring 2004 Exercises for CG and String Matching -------------------------------------------------- Working modulo q= 11, how many spurious hits does the Rabin-Karp matcher encounter in the text T = 3141592653589793 when looking for the pattern P = 26. How would you extend the Rabin-Karp method to handle the problem of searching a text string for an occurrence of any of a given set of k patterns? Show how to extend the Rabin-Karp method to handle the problem of looking for a given m ( m pattern in an n ( n array of characters. (The pattern may be shifted vertically or horizontally, but it may not be rotated.) The input is two strings of characters A = a1 a2 . . . an and B = b1 b2 . . . bn. Design an O(n) algorithm to determine whether B is a cyclic shift of A. In other words, the algorithm should determine whether there exists an index k, 1( k( n such that ai = b(k+i) mod n, for all i, 1( i( n. Let P be a set of n points in the plane. We define the depth of a point p in P as the number of convex hulls that need to be peeled for p to become a vertex of the convex hull. Design an O(n2) algorithm to find the depths of all points in P. Let A = a1 a2 (an and B= b1b2(bm be two strings of characters. Denote by A[i] the string aiai+1(an(namely he ith suffix of A). Let di be the minimal edit distance between B and A[i]. Design an O(n2) algorithm to find the minimum value of di be the edit distance between B and A[i]. Design an O(n2) algorithm to find the minimum value of di( among all 1(i(n).    e f h i w x | Z [ \ ַַַ֣֮֮֮hG}hIC6H*OJQJhG}hIC6OJQJhG}hICOJQJhIC6OJQJ jhICOJQJhICH*OJQJ jhICOJQJhICOJQJhIC5B* OJQJ\ph#hIChIC5B* OJQJ\ph3;no u  $ & Fa$gdIC $h^ha$gdIC $h^ha$gdIC      ; < A E Q T U V  jhG}hICOJQJ^JhG}hIC6H*OJQJ^J! jhG}hIC6OJQJ^JhG}hIC6H*OJQJ^JhG}hIC6OJQJ^JhG}hICOJQJ^J? h:ahG}hICOJQJ^JhICOJQJ^J 1h/ =!"#$%H`H ICNormal CJPJ_HaJmH nHsH tHDAD Default Paragraph FontRiR  Table Normal4 l4a (k(No List;nou@0@0p@0p@0p@0p@ 0p@ 0@ 0@ 0@ 0@ 0@0@0u@ 0b@ 0b@ 0b@ 0 @ 0  @0    vx}CD~ {}XZQS33333333 Mohan KumarbS R1J(hh^h`o(.^`o(. ^`hH. pLp^p`LhH. @ @ ^@ `hH. ^`hH. L^`LhH. ^`hH. ^`hH. PLP^P`LhH.R1bSl         ;C:aIC;@P@UnknownGz Times New Roman5Symbol3& z Arial;SimSun[SO;|i0Batang"qh  !243Q H)?;CCSE 5311 Spring 2004 Mohan Kumar Mohan Kumar  Oh+'0   < H T`hpxCSE 5311 Spring 20041SE  Mohan KumarohaohaNormalu Mohan Kumar1haMicrosoft Word 10.0@G@ȳ#@B#՜.+,0 hp  CSE@UTA A CSE 5311 Spring 2004 Title  !"#$'Root Entry F? g#)1Table WordDocument"SummaryInformation(DocumentSummaryInformation8CompObjj  FMicrosoft Word Document MSWordDocWord.Document.89qcse5311-old/desktop.ini010064400020320000200000000001621003507357300153160ustar00cse5311cse531100000400000024[.ShellClassInfo] IconFile=_vti_pvt/fpdbw.ico IconIndex=0 ConfirmFileOp=0 InfoTip=Stores your local web site cse5311-old/ha1sp04.doc010064400020320000200000000650001003507357300150150ustar00cse5311cse531100000400000024ࡱ> 02/5@ 0bjbj22 " XX vvvvvvv$KMMMMMMRMv  Mvvb vvvK K vv UC ^c'$x0y (vvvvv,LB 6 x , )MMnD nCSE 5311 Design and Analysis of Algorithms SPRING 2004 Department of Computer Science The University of Texas at Arlington Home Assignment I A d-ary heap is like a binary heap, but instead of 2 children, nodes have d children. How would you represent a d-ary heap in an array? What is the height of a d-ary heap of n elements in terms of n and d. Give an efficient implementation of EXTRACT-MAX. Analyze its running time in terms of d and n. Give an efficient implementation of INSERT. Analyze its running time in terms of d and n. If the input is d sequences of elements such that each sequence is already sorted, and there is a total of n elements. Design an O(n log d) algorithm to merge all the sequences into one sorted sequence. The input is a sequence of elements x1, x2, , xn, given one at a time. Design an O(n) algorithm to compute the kth smallest element using only O(k) memory spaces. The value of k is known ahead of time but the value of n is not known until the last element is seen. The input is a sequence of elements, given one at a time, inserted into a fixed memory space C. In the ith input step xi is put into C (and Cs previous content is erased). 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Assessment and General Notes

Statement of Ethics

References

Notes and Exercises

Homeworks

Lab Assignments

Quizzes

Exams

Research Papers

Important Dates

Announcements

CSE5311 Design and Analysis of Algorithms

 

Fall 2005

 

Instructor: Dr. Mohan Kumar

 

Course Description

 

Design and Analysis of Algorithms is THE most important basic course in any graduate computer science and engineering curriculum. It is vital for every computer science student to be fluent with algorithms and their analysis. ALGORITHMS ARE FUN; ALGORITHM ANALYSIS is a NECESSARY TOOL;  Students are encouraged to solve homework problems and discuss/solve problems in the class. Each student will be given one specific algorithm or problem to carry out an in-depth study. Typically, this course should be taken in the very first semester of your graduate study because algorithms are used in Networks, Operating Systems, Databases, and other (including advanced) courses.

Course Objectives

The objective of this course is to build a solid foundation of the most important fundamental subject in computer science. Creative thinking is essential to algorithm design. Algorithm analysis and verification demands sound mathematical acumen and programming skills.

Course Prerequisites

Data Structures (CSE 2320) and Theoretical Concepts in Computer Science and Engineering (CSE 3315) OR Equivalent.

Mode of Teaching

Mode of Teaching: The class meets twice a week (Mondays and Wednesdays 10:30 to 11:50 am). The Monday class will be of lecture type and the Wednesday class will be of tutorial type. Power point slides and other lecture material will be used on Mondays. At the end of each topic, students must attempt to solve exercise problems. Exercise problems can be found on the course web page and in the text book. All students are expected to work on these problems and may be required to solve these in the class on Wednesdays.

 

Instructor: Mohan Kumar, 309 Nedderman Hall

Email: kumar@cse.uta.edu

Office Hours:    Wednesdays 12 to 3:00 PM   

Phone: (817) 272-3610 or main office: (817) 272-3785;

WWW site: http://crystal.uta.edu/~kumar/cse5311

GTA: TBA 

GTAs Email: TBA

Office Hours:  TBA

Office: TBA.

Course Syllabus:

      Review of Asymptotic Analysis and Growth of Functions; Trees, Heaps, and Graphs; and Recurrences.

 

      Greedy Algorithms: Minimum spanning tree, Union-Find algorithms, Kruskal's Algorithm, Clustering, Huffman Codes, and Multiphase greedy algorithms.

 

      Dynamic Programming: Shortest paths, negative cycles, matrix chain multiplications, sequence alignment, RNA secondary structure, application examples.

 

    Network Flow: Maximum flow problem, Ford-Fulkerson algorithm, augmenting paths, Bipartite matching problem, disjoint paths and application problems.

 

      NP and Computational tractability: Polynomial time reductions; The Satisfiability problem; NP-Complete problems; and Extending limits of tractability.

      Approximation Algorithms, Local Search and Randomized Algorithms

 

 

 

Text book

           Algorithm Design
           by Jon Kleinberg, va Tardos

           Pearson Addison-Wesley

           ISBN 0-321-29535-8

References

      The Design and Analysis of Algorithms 1974

AV Aho, JE Hopcroft and JD Ullman, Addison-Wesley Publishing Company

      Introduction to Algorithms: A Creative Approach, Reprinted 1989

Udi Manber, Addison-Wesley Publishing Company

       Introduction to Algorithms, Second Edition, 2001

T Cormen, C E Leiserson, R L Rivest and C Stein McGraw Hill and MIT Press

      Graph Algorithms, 1979

Shimon Even, Computer Science Press

      Introduction to the Theory of Computation, 1992

Michael Sipser, PWS Publishing Company

      The Art of Computer Programming, Vols. 1 and 3

Knuth, Addison Wesley Publishing Company

Assessment

Course grades will be based on the following:

Class participation: ACTIVE Participation will prepare you well for Quizzes and Exams Students are expected to interact actively during lectures. All students are expected to solve homework problems and discuss solutions in the class.

 

Quizzes and class participation: 30%

There will be 2 quizzes.

 

The structure of the quizzes will be discussed in class, one week

at least prior to the quiz.

 

Quiz 1 (15%) : September 14, 2005

Quiz 2 (15%) :  November 18, 2005

 

Exams: 35%

There will be ONE exam. The questions will test your ability to write and analyze algorithms. The exam will NOT test your memory. The exam will be a comprehensive

open book/ notes, exam lasting for about 3-4 hours.

 

Exam:  October 31, 2005

 

Group Project/study : 35%

 

Students will have the option of doing a group study or group project.

Project problems will handed out at least one month prior to the expected date of completion. The students will be required to write programs and run experiments.

 

OR
 

Groups of students will be given a research problem to work on. Students are encouraged to come up with new and creative ideas.

 

A presentation of this work will  be on December 2, 2005.

 

 

Homework Assignments: No Grades awarded directly!

Missed Exams, Quizzes, and Makeup Work

If you miss an exam or quiz due to unavoidable circumstances (e.g., health), email the instructor for an appointment or meet with him during office hours. Do NOT ask for make up exams or other components if you missed an exam or a project due to travel (except when you are required to travel to represent the university or the department).

Attendance and Drop Policy

Attendance though not mandatory, is HIGHLY encouraged. Class participation is important to your grade in the 'Quizzes and Class Participation' component

Final Review Week

A period of five class days prior to the first day of final examinations in the long sessions shall be designated as Final Review Week. The purpose of this week is to allow students sufficient time to prepare for final examinations. During this week, there shall be no scheduled activities such as required field trips or performances; and no instructor shall assign any themes, research problems or exercises of similar scope that have a completion date during or following this week unless specified in the class syllabi. . During Final Review Week, an instructor shall not give any examinations constituting 10% or more of the final grade, except makeup tests and laboratory examinations. In addition, no instructor shall give any portion of the final examination during Final Review Week.

Americans With Disabilities Act

The University of Texas at Arlington is on record as being committed to both the spirit and letter of federal equal opportunity legislation; reference Public Law 93112 -- The Rehabilitation Act of 1973 as amended. With the passage of new federal legislation entitled Americans With Disabilities Act - (ADA), pursuant to section 504 of The Rehabilitation Act, there is renewed focus on providing this population with the same opportunities enjoyed by all citizens.

As a faculty member, I am required by law to provide "reasonable accommodation" to students with disabilities, so as not to discriminate on the basis of that disability. Student responsibility primarily rests with informing faculty at the beginning of the semester and in providing authorized documentation through designated administrative channels.

Academic Dishonesty

It is the philosophy of The University of Texas at Arlington that academic dishonesty is a completely unacceptable mode of conduct and will not be tolerated in any form. All persons involved in academic dishonesty will be disciplined in accordance with University regulations and procedures. Discipline may include suspension or expulsion from the University.

"Scholastic dishonesty includes but is not limited to cheating, plagiarism, collusion, the submission for credit of any work or materials that are attributable in whole or in part to another person, taking an examination for another person, any act designed to give unfair advantage to a student or the attempt to commit such acts." (Regents Rules and Regulations, Part One, Chapter VI, Section 3, Subsection 3.2, Subdivision 3.22)

Student Support Services Available

The University of Texas at Arlington supports a variety of student success programs to help you connect with the University and achieve academic success. These programs include learning assistance, developmental education, advising and mentoring, admission and transition, and federally funded programs. Students requiring assistance academically, personally, or socially should contact the Office of Student Success Programs at 817-272-6107 for more information and appropriate referrals.
 

 

 

   
rticipation
: ACTIVE Participation will prepare you well for Qucse5311-old/research_list.doc010064400020320000200000000620001011514475200164560ustar00cse5311cse531100000400000024ࡱ> -/,5@ 0 bjbj22 "XXlllllll (lnnnnnnR XnlnllRllll (ll( lNv(l0(e :e (llll(e l<0nn$ List of Research Papers for CSE 5311 Fall 2004 1-4: Major Work 5-6: Minor Work 1. Eppstein D. Offline Algorithms for Dynamic Minimum Spanning Tree Problems  INCLUDEPICTURE "http://www.sciencedirect.com/scidirimg/clear.gif" \* MERGEFORMATINET  HYPERLINK "http://www.sciencedirect.com/science?_ob=JournalURL&_cdi=6839&_auth=y&_acct=C000004378&_version=1&_urlVersion=0&_userid=108428&md5=024a96bbed47bb4b76d9ac116412ccd3" Journal of Algorithms, Volume 17, Issue 2, Pages 237-250 (September 1994). 2.Galil Z., Efficient algorithms for finding maximum matching in graphs, ACM Computing Surveys, Vol 18, March 1986, pgs. 23-38. 3. Johnson D.B. Efficient Algorithms for shortest paths in sparse networks, Journal of the ACM, Vol. 24, January 1977, pgs. 1-13. 4. Goldberg A.V., Tarjan, A new approach to the maximum flow problem, Journal of the ACM, vol. 35, October 1988, pgs. 921-940. 5. Incerpi J., and R. Sedgewick, Practical variations of shellsort, Information Processing Letters, Vol. 26, 1987, pgs. 37-43. 6. Hunt J.W., and Szymanski T.G., A fast algorithm for computing longest common subsequences, Communications of the ACM, Vol. 20, May 1977, pgs. 350-353. 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A[small] A[i]; 8. A[i] temp; 9. end V    `01?? 0 ,$ 0 Example : Given sequence 5 2 4 6 1 3 i=1 1 2 4 6 5 3 i=2 1 2 4 6 5 3 i=3 1 2 3 6 5 4 i=4 1 2 3 4 5 64H  0޽h ? ̙33!!___PPT10!+9Dp!' *= @B D+!' = @BA?%,( < +O%,( < +D_' (=%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D_' (=%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*2D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*2D_' (=%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*3M%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*3MD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*3MD_' (=%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*Mg%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*MgD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*MgD_' (=%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*g%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*gD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*gD_' (=%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(#/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+8+0+ +  vn(    0Ћf  Complexity: The statements 2,6,7,8, and 5 take O(1) or constant time. The outerloop 1-9 is executed n-1 times and the inner loop 3-5 is executed (n-i) times. The upper bound on the time taken by all iterations as i ranges from 1 to n-1 is given by, O(n2)x K   # lՋ1??p  1. for i 1 to n-1 2. small i; 3. for j i+1 to n 4. if A[j] < A[small] then 5. small j; 6. temp A[small]; 7. A[small] A[i]; 8. A[i] temp; 9. end&  = H  0޽h ? ̙33p    H( Y Hx H c $ڋ `  H H 0޽h ? ̙33f   L(  Lr L S    r L S D0    L 0 P @y  *A program is an expression of an algorithm in a programming language. *Choice of proper data models and hence data structures is important for expressing algorithms and implementation. Hs H L 0޽h ? ̙33y___PPT10Y+D=' *= @B +m    P$(  Pr P S *  `   r P S + Pp    H P 0޽h ? ̙33y___PPT10Y+D=' *= @B +   0T$( b  Tr T S 8    r T S |9    H T 0޽h ? ̙33`  x@( ʆ x  c $K  `}   l  0A ??@C &   `XY 1?? i,$ 0 bc = 4 n0 = 3.58 2    `xi 1??` f,$ 0 f(n), is O (g(n)) if there exist positive constants c and n0 such that 0 f(n) c g(n) for all n n0. o# p   @     H  0޽h ? 33f  ___PPT10` +9D ' *= @B D ' = @BA?%,( < +O%,( < +D_' (=%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D_' (=%(D' =%(D' =A@BBBB0B%(E' =4 B`BPB`B?<*%( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*+p+0+  ++0+  +d   P( T l  0A ??or  H  0޽h ? 3ffj   `X(  Xr X S     H X 0޽h ? ̙33   p\<(   \r \ S   @     \ 0  & *rO, W, and Q correspond (loosely) to   ,   , and  = . : H \ 0޽h ? ̙33   `$( v  `r ` S p p   r ` S      H ` 0޽h ? ̙33t   $d( v  d| d 0dȂ   SELECTION SORT Algorithm (Iterative method) Procedure SELECTION_SORT (A [1,& ,n]) Input : unsorted array A Output : Sorted array A 1. for i 1 to n-1 2. small i; 3. for j i+1 to n 4. if A[j] < A[small] then 5. small j; 6. temp A[small]; 7. A[small] A[i]; 8. A[i] temp; 9. end V H d 0޽h ? ̙33  rjl(  l l 0p   jRecursive Selection Sort Algorithm Given an array A[i, & ,n], selection sort picks the smallest element in the array and swaps it with A[i], then sorts the remainder A[i+1, & , n] recursively. Example : Given A [26, 93, 36, 76, 85, 09, 42, 64] Swap 09 with 23, A[1] = 09, A[2,& , 8] = [93,36,76,85,26,42,64] Swap 26 with 93, A[1,2]= [09,26]; A[3,& ,8] = [36,76,85,93,42,64] No swapping A[1,2,3] = [09,26,36]; A[4,& ,8] =[76,85,93,42,64] Swap 42 with 76, A[1,& ,4] =[09,26,36,42]; A[5,& ,8] = [85,93,76,64] Swap 64 with85, A[1,& ,5] =[09,26,36,42,64]; A[6,7,8] = [93,76,85] Swap 76 with 93, A[1,& ,6]=[09,26,36,42,64,76]; A[7,8] = [93,85] Swap 85 with 93, A[1,& ,7]=[09,26,36,42,64,76,85]; A[8] = 93 Sorted list : A[1,& ,8] = [09,26,36,42,64,76,85,93] d$%67IF>BtH l 0޽h ? ̙33^  p(  pf p <L 9F XProcedure RECURSIVE_SELECTION_SORT (A[1,& ,n],i,n) Input : Unsorted array A Output : Sorted array A while i < n do small i; for j i+1 to n if A[j] < A[small] then small j; temp A[small]; A[small] A[i]; A[i] temp; RECURSIVE_SELECTION_SORT(A,i+1,n) End - '2 '  -H p 0޽h ? ̙33   t(  t t <.   zLAnalysis of Recursive selection sort algorithm Basis: If i = n, then only the last element of the array needs to be sorted, takes Q (1) time. Therefore, T(1) = a, a constant Induction : if i < n, then, 1. we find the smallest element in A[i,& ,n], takes at most (n-1) steps swap the smallest element with A[i], one step recursively sort A[i+1, & , n], takes T(n-1) time Therefore, T(n) is given by, T(n) = T(n-1) + b. n (1) It is required to solve the recursive equation, T(1) = a; for n =1 T(n) = T(n-1) + b n; for n >1, where b is a constant _h6/M  [_h4H t 0޽h ? ̙33  x4(  x x <G s  T(n-1) = T(n-2) + (n-1)b (2) T(n-2) = T(n-3) + (n-2) b (3) . . . T(n-i) = T(n-(i+1)) + (n-i)b (4) Using (2) in (1) T(n) = T(n-2) + b [n+(n-1)] = T(n-3) + b [n+(n-1)+(n-2) = T(n-(n-1)) + b[n+(n-1)+(n-2) + . . . +(n-(n-2))] T(n) = O(n2) lii H x 0޽h ? ̙33  D<|( 3+  | | 0\T  <& Questions: What is an algorithm? Why should we study algorithms? Why should we evaluate running time of algorithms? What is a recursive function? What are the basic differences among O, W, and Q notations? Did you understand selection sort algorithm and its running time evaluation? Can you write pseudocode for selecting the largest element in a given array? Please write the algorithm now.  -!,C  CH | 0޽h ? ̙33  8Q(  8 8 0g p  hHeaps and Heapsort 2 u 8 0@n   kThis week Priority Trees Building Heaps Maintaining heaps Heapsort Algorithm Analysis of Heapsort Algorithm6 bq bP 0      8 0u p l*Further Reading Chapters 6 from Textbook &+*H 8 0޽h ? ̙33  <(  <x < c $`y     < 0| P0n GWhat is a priority queue? A priority queue is an abstract data type which consists of a set of elements. Each element of the set has an associated priority or key Priority is the value of the element or value of some component of an element: < 0  0 VExample : S : {(Brown, 20), (Gray, 22), (Green, 21)} priority based on name {(Brown, 20), (Green,21), (Gray, 22)} priority based on age Each element could be a record and the priority could be based on one of the fields of the record H 2 IH < 0޽h ? 3ff  f^@(  @x @ c $b     > @ 0 `  <A Student's record: Attributes : Name Age Sex Student No. Marks Values : John Brown 21 M 94XYZ23 75 Priority can be based on name, age, student number, or marks Operations performed on priority queues, inserting an element into the set finding and deleting from the set an element of highest priorityf-Y>)eH @ 0޽h ? 3ff  zrD (  Dx D c $4` p   R D 0 `  pPriority queues are implemented on partially ordered trees (POTs) POTs are labeled binary trees the labels of the nodes are elements with a priority the element stored at a node has at least as large a priority as the elements stored at the children of that node the element with the highest priority is at the root of the tree2B/!p,<%H D 0޽h ? 3ff  H(  Hx H c $ԭ    ` H c $A ??@P:  H H 0޽h ? 3ff8  Lx(  Lx L c $ @    L 0 ` `The heap is a data structure for implementing POT's Each node of the heap tree corresponds to an element of the array that stores the value in the node The tree is filled on all levels except possibly the lowest, which are filled from left to right up to a point. An array A that represents a heap is an object with two attributes length[A], the number of elements in the array and heap-size[A], the number of elements in the heap stored within the array A heap_size[A] length[A] .4 2,E: K .H L 0޽h ? 3ff  P,(  Px P c $|H @   t P 0 @ The heap comprises elements in locations up to heap-size[A] . A[1] is the root of the tree. Position 1 2 3 4 5 6 7 8 9 10 Value 24 21 19 13 14 3 10 2 7 11 Given node with index i, PARENT(i) is the index of parent of i;PARENT(i) = i/2 LEFT_CHILD(i) is the index of left child of i ; LEFT_CHILD(i) = 2i; RIGHT_CHILD(i) is the index of right child of i; and RIGHT_CHILD(i) = 2i +1.> /   !H P 0޽h ? 3ff   TD(  Tx T c $ @    T 0 ``L  ,\THE HEAP PROPERTY A[PARENT(i)] A[i] The heap is based on a binary tree The height of the heap (as a binary tree) is the number of edges on the longest simple downward path from the root to a leaf. The height of a heap with n nodes is O (log n). All basic operations on heaps run in O (log n) time.J#H T 0޽h ? 3ff  3+0::X(  X:2 X 3 1P a4 X # ~ < !4B X@ # 1@ F J X J< X # 5eB2 X 3 1*< X # 37<B  XB # 1<B  X # 1N )=u  X  JB2  X 3 1?=P<  X # R+B2 X 3 1Eu< X # *CB2 X 3 1)[< X # <B XB # 1T<B X # 15)N )=u X PJB2 X 3 1?=P< X # R+B2 X 3 1Eu< X # *CB2 X 3 1)[< X # <B XB # 1T<B X # 15)F J X pLX< X # 5eB2 X 3 1*<  X # 37<B !XB # 1<B "X # 1N )=u #X  JB2 $X 3 1?=P< %X # R+B2 &X 3 1Eu< 'X # *CB2 (X 3 1)[< )X # <B *XB # 1T<B +X # 15)N )=u ,X PJB2 -X 3 1?=P< .X # R+B2 /X 3 1Eu< 0X # *CB2 1X 3 1)[< 2X # <B 3XB # 1T<B 4X # 15)XB 5X 0Dp ^B 6X 6Dp ^B 7X 6Dp ^B 8X 6Dp   9X 0# P  20 21 22 23 2h 2  :X 0, @ p`  +#n=20+21+22+23 + . . . + 2h = 2h+1-1$ 2        H X 0޽h ? 3ffc   @\(  \~ \ s *;  `}    \ 0(= @  = HEAPIFY BUILD_HEAP HEAPSORT HEAP_EXTRACT_MAX HEAP_INSERT,><H \ 0޽h ? 3ff   l d P` (  `x ` c $E     ` 0TH  J 4rThe HEAPIFY algorithm checks the heap elements for violation of the heap property and restores heap property. Procedure HEAPIFY (A,i) Input: An array A and index i to the array. i =1 if we want to heapify the whole tree. Subtrees rooted at LEFT_CHILD(i) and RIGHT_CHILD(i) are heaps Output: The elements of array A forming subtree rooted at i satisfy the heap property.`sH 3>^( ` 0pc @ H41. l LEFT_CHILD (i); 2. r RIGHT_CHILD (i); 3. if l heap_size[A] and A[l] > A[i] 4. then largest l; 5. else largest i; 6. if r heap_size[A] and A[r] > A[largest] 7. then largest r; 8. if largest i 9. then exchange A[i] A[largest] 10. HEAPIFY (A,largest)   H ` 0޽h ? 3ffG  nf` d(  d d B CDEFb@@XSr;W?-$K\m~ &&;Rft}fVK^vHdK o   p&>&>11@%DSME+~X.0@                     `? Z d BoCDEF^==H{S7"}&J#0707QlP<vHb@okobNZn1}"'Egto}'Uwf=C?3=J3-4,0@                    `@` d c $A ??0   d c $dA ?  8 $D 0 d c $dA ?p  8 $D 0 d c $dA ? `8 $D 0 d c $dA ?  `8 $D 0  d 0<  `  9 LST; heap 2   d 0Ȕ ` 0  9 RST, heap 2 H d 0޽h ? 3ffy___PPT10Y+;^D=' *= @B D ' = @BA?%,( < +O%,( < +Dn' (=%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*dD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*dD' (=%(D' =%(D3' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =-o6Bbox(out)*<3<*dD' (=%(D' =%(D3' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =-o6Bbox(out)*<3<*dD' (=%(D' =%(D3' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =-o6Bbox(out)*<3<*d+z  *"ph(  h~ h 0 4 x7 24 19 21 14 03 10 02 13 11 24 7 19 21 14 03 10 02 13 11 24 21 19 07 14 03 10 02 13 11 24 21 19 13 14 03 10 02 07 11y ! h 0d  p *Running time of HEAPIFY Total running time = steps 1 & 9 + recursive call T (n) = Q (1) + T (n/2 ) Solving the recurrence, we get T (n) = O (log n) V 2SAH h 0޽h ? 3ff  lF(  lx l c $ܺ    h l 0` p Procedure BUILD_HEAP (A) Input : An array A of size n = length [A], heap_size[A] Output : A heap of size n 1. heap_size[A] length[A] 2. for i length[A]/2 downto 1 3. HEAPIFY(A,i)!0 % l <D 0 *p,$D  0 ~18 12 54 75 64 25 42 78 96 18 12 54 96 64 25 42 78 75 18 12 54 96 64 25 42 78 75 18 96 54 12 64 25 42 78 75 18 96 54 78 64 25 42 12 75 96 18 54 78 64 25 42 12 75 96 78 54 18 64 25 42 12 75 96 78 54 75 64 25 42 12 18   H l 0޽h ? 3ff___PPT10+CD' *= @B DD' = @BA?%,( < +O%,( < +D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*l%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*lD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*l+8+0+l  +;  IIp{(  p:2 p 3 1Pe4 p # ]>6 p 6 `> >18 :2 p 3 1 R 4 p # I   p 6 L  >12 @2 p C 1f 4  p # ]u+6   p 6@ au >54 2  p C  1 T  >96 :2  p 3 1 & 4  p #  R  p 6 $   >64 :2 p 3 1 i 4 p #  e  p 6  e  >25 :2 p 3 1  4 p #  L  p 6,     >42 :2 p 3 1  S 4 p # ( u   p 6 , u  >75 :2 p 3 1   4 p #  #  p 6%  #  >78 4B p@ # 14B p # 1&4B p@ # 1 4B p # 1` 4B p@ # 1@ 5 4B  p # 1K ^ 4B !p@ # 1 Z 4B "p # 1 ` :2 #p 3 1"u4 $p # mTF %p 6x- p >18 :2 &p 3 1zb4 'p # Yk . 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Each call to HEAPIFY takes O (log n) time 2. There are O (n) such calls 3. Therefore the running time is at most O( n logn) However the complexity of BUILD_HEAP is O(n) Proof : In an n element heap there are at most n/2h+1 nodes of height h The time required to heapify a subtree whose root is at a height h is O(h) (this was proved in the analysis for HEAPIFY) So the total time taken for BUILD_HEAP is given by, We know that Thus the running time of BUILD_HEAP is given by, O(n) >t 2L0-'^cH>z 0HH`  c $A ?@p   `  c $A ??l  t  `  c $A ?? @ `  c $A ?? l S  H  0޽h ? 3ff  x(  x  c $ @@p   X  0 PE  Procedure HEAPSORT(A) Input : Array A[1& n], n = length[A] Output : Sorted array A[1& n] 1. BUILD_HEAP[A] 2. for i length[A] down to 2 3. Exchange A[1] A[i] 4. heap_size[A] heap_size[A]-1; 5. HEAPIFY(A,1) Example : To be given in the lecture  2r(H  0޽h ? 3ff  tl(  x  c $$    L  0<& 0A  Running Time: Step 1 BUILD_HEAP takes O(n) time, Steps 2 to 5 : there are (n-1) calls to HEAPIFY which takes O(log n) time Therefore running time takes O (n log n) * 2H  0޽h ? 3ff"  b(  x  c $(1 @     0D4  `5 J Procedure HEAP_EXTRACT_MAX(A[1& n]) Input : heap(A) Output : The maximum element or root, heap (A[1& n-1]) 1. if heap_size[A] 1 2. max A[1]; 3. A[1] A[heap_size[A]]; 4. heap_size[A] heap_size[A]-1; 5. HEAPIFY(A,1) 6. return max Running Time : O (log n) time  2( 2 #'$H  0޽h ? 3ff  B: (  x  c $xI @     0L `Pp \Procedure HEAP_INSERT(A, key) Input : heap(A[1& n]), key - the new element Output : heap(A[1& n+1]) with k in the heap 1. heap_size[A] heap_size[A]+1; 2. i heap_size[A]; 3. while i > 1 and A[PARENT(i)] < key 4. A[i] A[PARENT(i)]; 5. i PARENT(i); 6. A[i] key Running Time : O (log n) time B. 2 &5'$H  0޽h ? 3ff8  0x(  ~  s *| `}     0~P`  ZWhat is a heap? What are the running times for heap insertion and deletion operations ? 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Examples Slide 6Slide 7Slide 8"Algorithms and Design of Programs Slide 10Asymptotic Notations O Notation Slide 13 Slide 14 Slide 15Presenting algorithms Slide 17 Slide 18 Slide 19 Slide 20 Slide 21 Slide 22 Slide 23Priority QueuesExamplePriority QueuesExampleHEAPSHEAPS (Contd)Heap Property Slide 31Heap Algorithms HEAPIFY Slide 34 Slide 35 BUILD_HEAP Slide 37 Slide 38 Slide 39 Slide 40 Slide 41Running time of Build_heapThe HEAPSORT Algorithm HEAPSORTHEAP_EXTRACT_MAX HEAP_INSERT Questions:  Fonts UsedDesign TemplateEmbedded OLE Servers Slide Titles/#_% Mohan KumarMohan Kumar  !"#$%&'()*+,-./012345678:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~      !"#$%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwxyz{|}~Root EntrydO)PicturespCurrent UserSummaryInformation(PowerPoint Document(9IDocumentSummaryInformation8Root EntrydO)ƤP@PicturespCurrent UserSummaryInformation( ՜.+,D՜.+,x     On-screen ShowCurtin Universityes,5/A :Times New RomanArialBatangGulimSymbol WingdingsTimesDefault DesignMicrosoft Graph 97 ChartMicrosoft Word PictureMicrosoft Equation 3.0-CSE5311 Design and Analysis of Algorithms Slide 2Slide 3What are Algorithms ? 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DWingdingsRoman?l?0?z[ 0|?0DSymbolgsRoman?l?0?z[ 0|?@ .  @n?" dd@  @@`` |     M  OkO,2$K0=W3 ; 0AAp3333333@g4GdGd?z[ 0"ppp@  ʚ;ʚ;<4dddd?|- 0?<4BdBd?|- 0? 0___PPT10 l___PPT9N/ 02 ?d+CSE5311 8M KUMAR CSE5311O =Quicksort and Mergesort  oThis week Quicksort algorithm Quicksort performance Quicksort analysis Mergesort algorithm Mergesort analysis ` dq 333d333333333 e  Quicksort   The worst case running time of Quicksort algorithm is O(n2) However, its expected running time is O ( n log n) Three-step divide-and-conquer process for sorting a subarray A[l..r] Divide : partition the array A[l..r] into two nonempty subarrays A[l..q] and A[q+1,r] such that each element of A[l..q] is less than or equal to each element of A[q+1,..,r] Conquer : sort the two subarrays A[l..q] and A[q+1..r] by recursive calls to Quicksort Combine : the subarrays are already sorted in place. No work is needed to combine them b^!9 t 3333#3333.3333:33 3333333 N33Example     Running Time of Quicksort&   Mergesort    Example      Runtime Complexity of Mergesort   Runtime Complexity of Mergesort     Questions   uWhat is a pivot element? Do you understand divide-and-conquer? What is running time if pivot element is at the center of the array? How is Mergesort different from Quicksort? Trace the algorithm with the help of an example? What is the use of TEMP array? Do you know why we execute steps 13 and 14 in the MERGE algorithm? What happens if i >q after the while loop (4-11)? HuB 2  Find applications for selection sort, heapsort, quicksort, and mergesort sorting algorithms? Which of these problems are suitable for different types of sorting operations? Identify applications for which each sorting algorithm works bestb&        ?Insertion Sort Counting Sort Find the Maximum Find the Minimum8The Odd-Even Mergesort problem, B 9 18 27 36 45 54 63 72 64 56 48 40 32 24 16 8 Bitonic Mergesort>C 0 1 Sorting Networks  !Shuffle-Exchange Network Week 6"!   Parallel Bitonic Sort Algorithm !!  Given Unsorted list :&   ` ̙33` ` ff3333f` 333MMM` f` f` 3>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> L(    6#U  P U  T Click to edit Master title style! !  0U   U  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0U  `` U  X*  0L  `  U  Z*  08L  `  U  Z*  F  ? Base"pp   H  0޽h ? ̙33 Default Design 0 0*(    0ش P    X*   0踐     Z* d  c $ ?    0p  @  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  6 `P   X*   6ǐ `   Z* H  0޽h ? ̙3380___PPT10.bq5 @(  l  C L  L  l  C -L    L    04F  K  3Further Reading Chapters 1 and 8 from Textbook H3 233H  0޽h ? ̙33  P ( v   l  C F    F  l  C xF p   F  H  0޽h ? ̙3354  `O(  l  C  F    F    0LF p` : 2   0p,$  0 m13 02 18 26 76 87 98 11 93 77 65 43 38 09 65 06 13 02 06 26 76 87 98 11 93 77 65 43 38 09 65 18 13 02 06 09 76 87 98 11 93 77 65 43 38 26 65 18 13 02 06 09 11 87 98 76 93 77 65 43 38 26 65 18 11 02 06 09 13 87 98 76 93 77 65 43 38 26 65 18 11 02 06 09 13 87 98 76 93 77 65 43 38 26 65 18 09 02 06 11 13 87 98 76 93 77 65 43 38 26 65 18 09 02 06 11 13 87 98 76 93 77 65 43 38 26 65 18 06 02 09 11 13 87 98 76 93 77 65 43 38 26 65 18 06 02 09 11 13 87 98 76 93 77 65 43 38 26 65 18 02 06 09 11 13 87 98 76 93 77 65 43 38 26 65 18 02 06 09 11 13 87 18 76 93 77 65 43 38 26 65 98 02 06 09 11 13 87 18 76 65 77 65 43 38 26 93 98 02 06 09 11 13 26 18 76 65 77 65 43 38 87 93 98 02 06 09 11 13 18 26 76 65 77 65 43 38 87 93 98 02 06 09 11 13 18 26 76 65 38 65 43 77 87 93 98 02 06 09 11 13 18 26 43 65 38 65 76 77 87 93 98 02 06 09 11 13 18 26 43 38 65 65 76 77 87 93 98 02 06 09 11 13 18 26 38 43 65 65 76 77 87 93 98 02 06 09 11 13 18 26 38 43 65 65 76 77 87 93 98  P 33< 3333!3333! 33f3331 3333f3331 f3f3331  f3f3331  f3f3f3331 f3f3f3331 f3f3f3f3f333( f3f3f3f3f333 f3f3f3f3f333 f3f3f3f3f333 f3f3f3f3f3f333f3f3  f3f3f3f3f3f333f3f3f3f3f3f3f3f333f3f3f3f3f3f3f3f3f333f3f3f3f3f3f3f3f3f333f3f3f3f3f3f3f3f3f3f3f333f3f3f3f3f3f3f3 f3H  0޽h ? ̙33___PPT10+ UD' A= @B D]' = @BA?%,( < +O%,( < +D' >=%(DG' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*b%(D' K=+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*bD' K=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*bD' >=%(DG' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*b%(D' K=+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*bD' K=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*bD' >=%(DG' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' K=+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*%D' K=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*%D' >=%(DG' =%(D' =A@BBBB0B@B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' K=+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*%D' K=+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*%+8+0+:  +g  p (    0m  ZProcedure Quicksort(A,l,r) Input :Unsorted Array (A,l,r) Output : Sorted subarray A(0..r) To sort the entire array A, l = 1 and r = length[A] if l < r then q PARTITION(A,l,r) QUICKSORT(A,l,q-1) QUICKSORT(A,q+1,r) >h 2    4  & 0HH8  P   P Z  s *  P Z  s *0  Z  s * 0    s *`:  p   ;X 2   0ߞ p   ^ y X4 2   0ў p `  d y X4 2H  0޽h ? ̙33      9 ( 0Wd >  0P0  Procedure PARTITION(A,l,r) Input : Array A(l .. r) Output : A and q such that A[i] A[q] for all i q and A[j] > A[q] for all j > q. x A[l]; i l; j r; while i < j do while A[i] x and i r do i i +1; while A[j] > x and j l do j j -1; if i< j then exchange A[i] A[j]; q j; exchange A[l] A[q];     %    F  P    `0Z  s *  P Z  s *0  Z  s * 0   s *  p   ;X 2   0p  p   ^ y X4 2   0X% p `  d y X4 2H  0޽h ? ̙33  | (   l  C F    F    0?`   T(n) = n-1 + T(i-1) + T(n-i) It takes n-1 comparisons for the partition Then we sort smaller sequences of size i-1 and n-i Each element has the same probability of being selected as the pivot, The average running time is given by, Detailed Analysis in the Classx! 2`   c $A ??@   H  0޽h ? ̙33~  .&$(  $l $ C DQ @    $ 0W Like Quicksort, Mergesort algorithm also is based on the divide-and conquer principle. Divide: This step computes the middle of the array, takes constant time, Q(1) Conquer : Two subproblems, each of size n/2 are recursively solved. Each subproblem contributes 2T(n/2) to the running time. Combine: Two sorted sequences are merged, this takes Q(n) time o 2aA z . H $ 0޽h ? ̙33H  O G ( (  (l ( C  k    3 ( 0Ȍ@c,$  0 13 02 18 26 76 87 98 11 93 77 65 43 38 09 65 06 13 02 18 26 76 87 98 11 93 77 65 43 38 09 65 06 13 02 18 26 76 87 98 11 93 77 65 43 38 09 65 06 02 13 18 26 76 87 98 11 93 77 65 43 38 09 65 06 02 13 18 26 76 87 98 11 93 77 65 43 38 09 65 06 02 13 18 26 76 87 98 11 93 77 65 43 38 09 65 06 02 13 18 26 76 87 98 11 93 77 65 43 38 09 65 06 02 13 18 26 76 87 98 11 93 77 65 43 38 09 65 06 02 13 18 26 76 87 11 98 93 77 65 43 38 09 65 06 02 13 18 26 11 76 87 98 93 77 65 43 38 09 65 06 02 11 13 18 26 76 87 98 93 77 65 43 38 09 65 06 02 11 13 18 26 76 87 98 06 09 38 43 65 65 77 99 02 06 09 11 13 18 26 38 43 65 65 76 77 87 93 98  201G G3G3 3G33 3133 3933 3 3133 3 33133 3 3313 33,3 33,0303H ( 0޽h ? ̙33)<!<___PPT10<+}?PD;' A= @B Dh;' = @BA?%,( < +O%,( < +D{' >=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(a%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(aD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(aD{' >=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(a%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(aD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(aD{' >=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(#%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(#D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(#D{' >=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(#%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(#D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(#D{' >=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(D{' >=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(F%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(FD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(FD{' >=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(F%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(FD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(FD{' >=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(D{' >=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(i%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(iD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(iD{' >=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(i%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(iD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(iD{' >=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(+%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(+D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(+D{' >=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(+%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(+D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(+D{' >=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*(+8+0+( +\   ,( v  ,d , 0$p3 ~Procedure MERGESORT(A,l,r) Input : A an array in the range 1 to n. Output: Sorted array A. if l < r then q (l+r )/2 ; MERGESORT(A,l,q) MERGESORT(A,q+1, r) MERGE (A,l,q,r) j[d 2D  H , 0޽h ? ̙33  R J 0 (  0 0 04  J Procedure MERGE(A,l,q,r) Inputs: two sorted subarrays A(l, q) and A(q+1, r) Output : Merged and sorted array A(l, r) i l; j q+1; k 0; while (i q) and ( j r) do k k+1; if A[i] A[j] then TEMP[k] A[i]; i i +1; else TEMP[k] A[j]; j j +1; if j > r then for t 0 to q  i do A[r-t] A[q-t]; for t 0 to k-1 do A[l +t] TEMP[t]; w 2     H 0 0޽h ? ̙33(  ''HL8B'(   8l 8 C 07    R 8 s *p 0: 8 3 o$ 8 6|9D =   8 6 T  @   8 6̚  =n :  8 3 o}  8 6- A "  8 6h An/2"  8 6`r-Z H n :  8 3 o 8 6H] A " 8 68 ? An/2" 8 6H  =n  8 zBgCDEFgbg @`C  8 zBfCDEFbf @`1 : 8 3 od 8 6G An/4" 8 6 J   An/2": 8 3 om 8 6u[ An/4" 8 6w   An/2" 8 6l#> W =n  8 zBCDEFz  @` 8 zBECDEF ;E @`x: 8 3 o l 8 6(  An/4" 8 6-   An/2"  8 61> XW =n : !8 3 ou "8 66"h An/4" #8 6@< "  An/2" $8 6A> "W =n  %8 zBCDEF{  @` &8 zBDCDEF ;D @`8 'Y  *8'Y B '8 3 'Y B (8 3 '- YZ B )8 3 ' Y 8 \  .8\ B +8 3 \ B ,8 3 \F r  -8 zB7CDEF272 @`\  @ /8 C o: 6k 08 681  W =   18 61  ! g  @   28 6( 1    =n @ 38 C o: k  48 61  ;o  A " 58 61   o  An/2"@ 78 C o. Z  88 6h1 V Ujo  A " 98 6D1  Lc  An/2" ;8 zBkCDEFfkf @`_ U J  <8 zBfCDEFfb @`J C J @ =8 C oN v{  >8 61    An/4" ?8 6#1 . )G  An/2"@ @8 C oN { {  A8 60(1  'm  An/4" B8 6(21 . 'G  An/2" C8 zBCDEF{ { @`j :  D8 zB@CDEF @7 @`j B @ E8 C oN ~{  F8 661    An/4" G8 6:1 . 1G  An/2"@ H8 C oN {  I8 6 @1  /u  An/4" J8 6D1 . /G  An/2" K8 zBCDEF{ { @`j :  L8 zB?CDEF ?6 @`j B H 8 0޽h ? ̙33  <Z(  <l < C M1    1   < 0+1 @ @ N.T(n) = 2 T(n/2) + Q(n) T(n/2) = 2 T(n/4) + n/2 T(n/4) = 2 T(n/8) + n/4 T(n) = 2{2 T(n/4) + n/2} + n = 4 T(n/4) + 2 n T(n) = 23 T(n/23) + 3n . . . T(n) = 2k T(n/2k) +k n If n = 2k then , k = log n Therefore, T(n) = 2k T(1) + n log n = n Q(1) + n log n T(n) = O(n log n)  2j     &  H < 0޽h ? ̙33  @(  @l @ C _1    1  l @ C `1 p 1  H @ 0޽h ? ̙33j  D( v  Dr D S g1  1  H D 0޽h ? ̙33$  |$(  |r | S %L P  L  r | S &L  L  H | 0޽h ? ̙3380___PPT10.dq!P     T (  Tx T c $j1    1   T 0\P05 RPorcedure PARALLEL ODD-EVEN MERGE SORT 1. for i 1 to N/2 ; N = number of data elements 2. for all PEj such that j is odd and 1 j N-1 3. if [PEj].a > [PEj+1].a 4. swap([PEj].a,[PEj+1].a) 5. for all PEj such that j is even and 1 j N-1 6. if [PEj].a > [PEj+1].a 7. swap([PEj].a,[PEj+1].a) } S 2 (    k T  pџ@  51k T  ԟ3 52k T  03G6 53q T # 8Q @` 5nLB T c $D H T 0޽h ? ̙33V     P` (  `x ` c $Drp     ` 0LIo @  ~A bitonic sequence is a sequence of numbers, a0,a1, . . ., an-1 such that for 0 i n-1 a0,a1, . . ., ai is in ascending order and ai+1,ai+2, . . ., an-1 is in descending order let, n = 2k, and i = n/2 -1. a0a1a2, . . ., an/2-2 an/2-1 . . . and an/2 an/2+1 an/2+2 an/2+3 . . . an-2 an-1 Consider, min (a0,an/2), min (a1,an/2+1), . . ., min(an/2-1,an-1) max(a0,an/2), max(a1,an/2+1), . . . , max(an/2-1,an-1) Both are bitonic sequences!! d/            %                            >`\H ` 0޽h ? ̙33  t%(  tx t c $o @  o   t 0$o   5+$dB t <DodB t <DodB t <Do dB t <Do  t <o ` 1a  t <Qo @` 9b  t <o `  @min(a,b)   t <Ho f  @max(a,b)   t 0`o J 5-$dB  t <Do dB t <Do dB t <DoJ*dB t <DoJ* t <o `  1a t <$o @ ` 9b t <o  @max(a,b)  t <0o f  @min(a,b) h t <o   J JCompare a0 with an/2 a1 with an/2+1 . . . an/2-1 with a n-1 K  t <o  x ^Compare data elements i and 2k-1+i and put the min in position 2i and the max in position 2i+1*_? t <o  k5compare_exchange_min(j); compare_exchange_max(j); 6K6H t 0޽h ? ̙33X$  $$NNx#(  xx x c $o p  o   x <o   = 87 75 63 31 L @p x# @ N @p x @pN @  x @ Z x s *@Z x s *` @ N @   x @ @pZ  x s *@Z  x s *` @ lB  x <Do@@lB  x <Do x 00o pF M+  x 0(o p& M+  x 0o p  M+  x 0o  p  M+ dB x <Do pP dB x <DopPdB x <DopPdB x <DopPdB x <DopPdB x <Do&pP&dB x <Do pP dB x <Do pP L   x#  fB x 6Do  lB x <DofB xB 6Do L   x#  fB x 6Do  lB  x <DofB !xB 6Do L   "x#   fB #x 6Do  lB $x <DofB %xB 6Do L   &x# fB 'x 6Do  lB (x <DofB )xB 6Do L   *x# fB +x 6Do  lB ,x <DofB -xB 6Do L   .x#  fB /x 6Do  lB 0x <DofB 1xB 6Do L  @ 2x# &@ fB 3x 6DjJ `B 4x 0D p `B 5x 0D `B 6x 0D ` `B 7x 0D` P `B 8x 0D@ @`B 9x 0D p `B :x 0D P L  @ ;x# &p  fB x 0D `B ?x 0D ` `B @x 0D` P `B Ax 0D@ @`B Bx 0D p `B Cx 0D P  Dx <o 0 + Ex <8o  + Fx <U   + Gx <`U    + Hx 0U  p F 1+ Ix 0hU  p & 1+ Jx 0U  p   1+ Kx 00L  p  1+ Lx <41L v  G31 45 63 67 75 78 87 89 Mx <8L R G 45 67 78 89 x  Nx < 7L  *  _#n/2 switches per stage log n stages $#H x 0޽h ? ̙33Zj  4 4`jjd3(  dF I @ d I @ d s *I \,$D 0 d s *@ S,$D 0 d s *6 I ,$D 0 d s *- @,$D 0 d 6(OL f3,$D 0 + d 6RL f3H-,$D  0 +  d 6UL f3Ht - ,$D  0 +  d 6YL f3Hl - ,$D  0 +Zz  I @  d I  @,$D 0  p  d#  I @,$D  0N @   d  Z d s *@Z d s *` @ N @  d @ pZ d s *@Z d s *` @  d 6l_L   ,$D  0 + d 6bL A |' ,$D 0 + d 6DfL A t ' ,$D 0 + d 6iL  ,$D 0 +Zz 0'' d 0'',$D 0  p d# 0'',$D 0N @  d  Z d s *@Z d s *` @ N @  d @ pZ d s *@Z d s *` @  d 6|oL +,$D 0 +  d 6rL |k,$D 0 + !d 6LvL t k ,$D 0 + "d 6yL l k ,$D 0 +"z G #d G,$D 0H   @ $d# G,$D 0fB %d 6Do  fB &d 6Do  fB 'd 6Do  fB (d 6Do @ @H   @ )d# Gt ,$D 0fB *d 6Do  fB +d 6Do  fB ,d 6Do  fB -d 6Do @ @ .d 0~L W,$D 0 U24 42 83 98 93 68 28 04"    /d# ,$D 0T   @ 0d#   @fB 1d 6Do  fB 2d 6Do  fB 3d 6Do  fB 4d 6Do @ @T   @ 5d#  `  fB 6d 6Do  fB 7d 6Do  fB 8d 6Do  fB 9d 6Do @ @`B :d 0D  `B ;d 0D  ` `B d 0D  `B ?d 0D `B @d 0D @ 0 `B Ad 0D     Bd#   ,$D 0T   @ Cd#   @fB Dd 6Do  fB Ed 6Do  fB Fd 6Do  fB Gd 6Do @ @T   @ Hd#  `  fB Id 6Do  fB Jd 6Do  fB Kd 6Do  fB Ld 6Do @ @`B Md 0D  `B Nd 0D  ` `B Od 0D  `B Pd 0D @ `B Qd 0D  `B Rd 0D `B Sd 0D @ 0 `B Td 0D     Ud#  ,$D 0T   @ Vd#   @fB Wd 6Do  fB Xd 6Do  fB Yd 6Do  fB Zd 6Do @ @T   @ [d#  `  fB \d 6Do  fB ]d 6Do  fB ^d 6Do  fB _d 6Do @ @`B `d 0D  `B ad 0D  ` `B bd 0D  `B cd 0D @ `B dd 0D  `B ed 0D `B fd 0D @ 0 `B gd 0D  hd 0L  x ,$D 0 U24 93 42 68 28 83 04 98" id 0L  W,$ 0 U24 28 83 93 04 42 68 98" jd 0L G,$D  0 U04 24 28 42 68 83 93 98"H d 0޽h ? 33f55___PPT105+wD5' A= @B D4' = @BA?%,( < +O%,( < +D{' >=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*hd%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*hdD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*hdDn' >=%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Bd%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*BdD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*BdDn' >=%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* d%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* dD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* dD' >=%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*id%(D' =-o6Bbox(out)*<3<*idD' >=%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*id%(D' =-o6Bbox(out)*<3<*idD' >=%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*id %(D' =-o6Bbox(out)*<3<*id D' >=%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*id %(D' =-o6Bbox(out)*<3<*id D' >=%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*id %(D' =-o6Bbox(out)*<3<*id D' >=%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*id%(D' =-o6Bbox(out)*<3<*idD' >=%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*id%(D' =-o6Bbox(out)*<3<*idD' >=%(D' =%(D@' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*id%(D' =-o6Bbox(out)*<3<*idD' >=%(D' =%(D3' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*Ud%(D' =-o6Bbox(out)*<3<*UdDn' >=%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*dD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*dDn' >=%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#d%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*#dD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*#dD' >=%(D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*jd%(D' =-m6Bbox(in)*<3<*jd++0+hdL  ++0+idL  ++0+jdL  +  ph"(  hx h c $(L    L  j h 0L p$ 41. Compare [PE(i)].a with [PE(i+2k-1)].a PE[2i] MIN { [PE(i)].a, [PE(i+2k-1)].a} PE[2i+1] MAX {[PE(i)].a, [PE(i+2k-1)].a} 2. Compare [PE(i)].a with [PE(i+2k-1)].a PE[2i] MIN { [PE(i)].a, [PE(i+2k-1)].a} PE[2i+1] MAX {[PE(i)].a, [PE(i+2k-2)].a} . . . k. Compare [PE(i)].a with [PE(i+2k-1)].a and PE[2i] MIN { [PE(i)].a, [PE(i+2k-1)].a} PE[2i+1] MAX {[PE(i)].a, [PE(i+2k-2)].a} for j = 1 to k Compare [PE(i)].a with [PE(i+2k-1)].a PE[2i] MIN { [PE(i)].a, [PE(i+2k-1)].a} PE[2i+1] MAX {[PE(i)].a, [PE(i+2k-1)].a} f3   f3    f3   f3   H h 0޽h ? ̙33  bZl(  lx l c $L   L  : l 0RF 0 B 98 24 42 83 68 04 93 28 98 24 42 and 83 in ascending order and 68 04 93 and 28 in descending order 98 24 42 83 68 04 93 28 98 24 42 83 68 04 93 28 24 98 83 42 68 04 28 93 24 42 83 98 68 93 28 04 24 42 83 98 93 68 28 04 |B 2! ""H l 0޽h ? ̙33N MMp0 =M(  p p 0F Pp \98 24 42 83 68 04 93 28"L  p p# P@P N @  p  Z p s *@Z p s *` @ N @  p @ pZ  p s *@Z  p s *` @   p <F Z -  p <F  Z  -  p <(F    +^B p 6D @^B p 6D@ @@^B p 6D @ ^B p 6D` @` ^B p 6D @ ^B p 6D @ L  L p#  L p 6F fL 3+`B p 0D @`B p 0D @ p 6F   U98 24 42 83 68 04 93 28"x p 6P : 0@z  Q24 98 83 42 68 04 28 93L  p p# PpP N @  p  Z p s *@Z p s *` @ N @  p @ pZ  p s *@Z !p s *` @  "p <: f3 3+ #p <: f3 3+ $p <$: f3`   3- %p <: f3   3-L  p &p# P P N @  'p  Z (p s *@Z )p s *` @ N @  *p @ pZ +p s *@Z ,p s *` @  -p <\: f3  3+ .p <: f3  3+ /p <<: f3`  3- 0p <: f3  3-dL   1p#  fB 2p 6Do@@fB 3p 6DofB 4p 6Do` p` fB 5p 6Do  fB 6p 6Do  fB 7p 6Do  fB 8p 6DopfB 9p 6Do`B :p 0Dp`B ;p 0D p `B p#  fB ?p 6Do@@fB @p 6DofB Ap 6Do` p` fB Bp 6Do  fB Cp 6Do  fB Dp 6Do  fB Ep 6DopfB Fp 6Do`B Gp 0Dp`B Hp 0D p `B Ip 0D p `B Jp 0Dp Kp 6 : 0pz  Q24 83 42 98 68 28 93 04X Mp 0P P X Np 0 P sX Op 0- P X Pp 0 P P  Qp <p: f3  3+ Rp <: f3 0  F 3+ Sp <(: f3` 0   3+ Tp <: f3 0   3+L  p Up# PP N @  Vp  Z Wp s *@Z Xp s *` @ N @  Yp @ pZ Zp s *@Z [p s *` @  \p <: f3&  + ]p <: f3`F + ^p <t: f3` `F  + _p <: f3 0&  +L  p `p# P`P N @  ap  Z bp s *@Z cp s *` @ N @  dp @ pZ ep s *@Z fp s *` @  gp <f3l + hp <<f3 + ip <xf3`   + jp <ĺf3   + L   @ kp# @fB lp 6Do  fB mp 6Do  fB np 6Do  fB op 6Do @ @ L   @ pp# `  fB qp 6Do  fB rp 6Do  fB sp 6Do  fB tp 6Do @ @ up 6d0P z  Q24 42 83 98 93 68 28 04L    vp#  T   @ wp#   @fB xp 6Do  fB yp 6Do  fB zp 6Do  fB {p 6Do @ @T   @ |p#  `  fB }p 6Do  fB ~p 6Do  fB p 6Do  fB p 6Do @ @`B p 0D  `B p 0D  ` `B p 0D  `B p 0D @ `B p 0D  `B p 0D `B p 0D @ 0 `B p 0D L    p# P  T   @ p#   @fB p 6Do  fB p 6Do  fB p 6Do  fB p 6Do @ @T   @ p#  `  fB p 6Do  fB p 6Do  fB p 6Do  fB p 6Do @ @`B p 0D  `B p 0D  ` `B p 0D  `B p 0D @ `B p 0D  `B p 0D `B p 0D @ 0 `B p 0D L    p# 0 T   @ p#   @fB p 6Do  fB p 6Do  fB p 6Do  fB p 6Do @ @T   @ p#  `  fB p 6Do  fB p 6Do  fB p 6Do  fB p 6Do @ @`B p 0D  `B p 0D  ` `B p 0D  `B p 0D @ `B p 0D  `B p 0D `B p 0D @ 0 `B p 0D  p 6`Ӗ0P z  Q24 93 42 68 28 83 04 98 p 6ؖ0J  Q24 28 83 93 04 42 68 98 p 6Ֆ`  Q04 24 28 42 68 83 93 98H p 0޽h ? 33fy___PPT10Y+D=' A= @B +xKh\Und23IlLf>I6 ")EmpS0v|f$V2N\_.Dn\,n.AɌwi3K%qx=۽zoA 65S(! 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(&;6<5=.) # #:7 , 4~   E7IqwAziW-V-@ !"#X$a%&' ()*+, X?2$t=^\0r$2$0 ]b=9 r 0e0e    A A1 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| f3f333@f38 g42d2dv 0:lppN<4BdBd` 0L2<4!d!d` 0L2g4?d?dv 0p,p ʚ;ʚ;<4dddd{ 0___PPT10DTimes New Roman((0(z[ 0(DArialNew Roman((0(z[ 0(" DSimSunew Roman((0(z[ 0( b___PPT9D/ 0(K?V- Kumar 0CSE5311 CSE@UTAO =-18Weeks 3 and 4 Graph Algorithms and Maximum Flow Networks29$$2Further Reading Chapter 22 .. 26 from Textbook 312Graph Preliminaries$  3 Konigsberg bridges     4Graph Terminologies  A Graph consists of a set 'V' of vertices (or nodes) and a set 'E' of edges (or links). A graph can be directed or undirected. Edges in a directed graph are ordered pairs. The order between the two vertices is important. Example: (S,P) is an ordered pair because the edge starts at S and terminates at P. The edge is unidirectional Edges of an undirected graph form unordered pairs. A multigraph is a graph with possibly several edges between the same pair of vertices. Graphs that are not multigraphs are called simple graphs.1 e1o3  5Graph Terminologies (Contd) 6Graph Terminologies  7Paths and Cycles 8Paths and Cycles 9 L Spanning Tree: Connectivity  ;Forest =Graph Representations( >Graph Representations ?Depth-first search  @DFS ADFS NDFSB  CBFS DBFS  EFConnected Components of a Graph  H I DFS Forest   J DFS ForestK Questions Do you know the difference between a simple graph and a multiple graph? What is an adjacency matrix ? What is a Hamiltonian path? What is an Euler path? Given a graph, can you find the Hamiltonian and Eulerian paths? Given a graph, can you perform DFS and BFS traversals? What is the difference between a cycle and a path? What are the complexities of basic operations on stacks and queues? Give proof. ,PMinimum-Cost Spanning Trees$Consider a network of computers connected through bidirectional links. Each link is associated with a positive cost: the cost of sending a message on each link. This network can be represented by an undirected graph with positive costs on each edge. In bidirectional networks we can assume that the cost of sending a message on link does not depend on the direction. Suppose we want to broadcast a message to all the computers from an arbitrary computer. The cost of the broadcast is the sum of the costs of links used to forward the message. r%Zub*fb f3bH1     QMinimum-Cost Spanning Trees$Find a fixed connected subgraph, containing all the vertices such that the sum of the costs of the edges in the subgraph is minimum. This subgraph is a tree as it does not contain any cycles. Such a tree is called the spanning tree since it spans the entire graph G. A given graph may have more than one spanning tree The minimum-cost spanning tree (MCST) is one whose edge weights add up to the least among all the spanning treesPZ ]SRS z5 qRMCSTSMCSTThe Problem: Given an undirected connected weighted graph G =(V,E), find a spanning tree T of G of minimum cost. Greedy Algorithm for finding the Minimum Spanning Tree of a Graph G =(V,E) The algorithm is also called Kruskal's algorithm. At each step of the algorithm , one of several possible choices must be made, The greedy strategy: make the choice that is the best at the momentqZZKZ3ZZ bK   TKruskal's Algorithm$  Procedure MCST_G(V,E) (Kruskal's Algorithm) Input: An undirected graph G(V,E) with a cost function c on the edges Output: T the minimum cost spanning tree for G T 0; VS 0; for each vertex v V do VS = VS {v}; sort the edges of E in nondecreasing order of weight while |VS| > 1 do choose (v,w) an edge E of lowest cost; delete (v,w) from E; if v and w are in different sets W1 and W2 in VS do W1 = W1 W2; VS = VS - W2; T T (v,w); return TP  @+;G/ :  UMCSTThe algorithm maintains a collection VS of disjoint sets of vertices Each set W in VS represents a connected set of vertices forming a spanning tree Initially, each vertex is in a set by itself in VS Edges are chosen from E in order of increasing cost, we consider each edge (v, w) in turn; v, w V. If v and w are already in the same set (say W) of VS, we discard the edge If v and w are in distinct sets W1 and W2 (meaning v and/or w not in T) we merge W1 with W2 and add (v, w) to T.,Z)VMCSTPConsider the example graph shown earlier, The edges in nondecreasing order [(A,D),1],[(C,D),1],[(C,F),2],[(E,F),2],[(A,F),3],[(A,B),3], [(B,E),4],[(D,E),5],[(B,C),6] EdgeActionSets in VSSpanning Tree, T =[{A},{B},{C},{D},{E},{F}]{0}(A,D)merge [{A,D}, {B},{C}, {E}, {F}] {(A,D)} (C,D) merge [{A,C,D}, {B}, {E}, {F}] {(A,D), (C,D)} (C,F) merge [{A,C,D,F},{B},{E}]{(A,D),(C,D), (C,F)} (E,F) merge [{A,C,D,E,F},{B}]{(A,D),(C,D), (C,F),(E,F)}(A,F) reject [{A,C,D,E,F},{B}]{(A,D),(C,D), (C,F), (E,F)}(A,B) merge [{A,B,C,D,E,F}]{(A,D),(A,B),(C,D), (C,F),(E,F)}(B,E) reject (D,E) reject (B,C) reject"Q!pP(Q  'W Complexity |Steps 1 thru 4 take time O (V) Step 5 sorts the edges in nondecreasing order in O (E log E ) time Steps 6 through 13 take O (E) time The total time for the algorithm is therefore given by O (E log E) The edges can be maintained in a heap data structure with the property, E[PARENT(i)] E[i] remember, this property is the opposite of the one used in the heapsort algorithm earlier during Week 2. This property can be used to sort data elements in nonincreasing order. Construct a heap of the edge weights, the edge with lowest cost is at the root During each step of edge removal, delete the root (minimum element) from the heap and rearrange the heap. The use of heap data structure reduces the time taken because at every step we are only picking up the minimum or root element rather than sorting the edge weights.X?Pj: @X mV  ` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> (    6p "8&}  T Click to edit Master title style! !$  0  "8&  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     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'@`  kBB  ` 3 >V/WHBB  ` 3 >/HBB  ` 3 >q3BB  ` 3 >y<BB ` 3 >E qF )BB ` 3 > } 6BB ` 3 >!BB ` 3 >!BB ` 3 >C BB ` 3 >nCoBB ` 3 > L BB ` 3 >X XY BB ` 3 > BB ` 3 >  ` 6L5 EA( ` 6Hi  CB&N ` S >) ` 6 ^  EC(N ` S >   ` 6$$   ED( !` <(`Z  pKonigsberg bridges(   $` 6-rC 9BF2 %` S 333jJg:F2 &` S 333jJ#jfF2 '` S 333jJJR )` B@1jJ`Z 9A +` 64 9C -` 6 8+d 9D:B .` 3 jJM :B /` 3 jJ;c]:B 0` 3 jJ03DF2 1` S 333jJ+:" 2` BtCDELFTjJ&Kn 0C[u5HWdm=setsn'(@3* 3` BuCDEPFXjJuN*|gnOU5@-tO' ),@F 4` B{CDEDFLjJ'LqpV 9$:M]jsYx-{#$@E* 5` B{CDEDFLjJ{T/ #5KeoWA.8b#$@0 ` 0= Z IThe town of Konigsberg (now Kaliningrad) lay on the banks and on two islands of the Pregel river. The city was connected by 7 bridges. The puzzle (as encountered by Leonhard Euler in 1736) : Whether it was possible to start walking from anywhere in town and return to the starting point by crossing all bridges exactly once. 8JH ` 03޽h ? ̙33y___PPT10Y+D=' *= @B +  dH( h  d~ d s *    d  6r   H d 03޽h ? ̙33y___PPT10Y+D=' *= @B +!  H@6Hh0 ( t h~ h s *C    h 68 <  MG1 :Undirected Graph F    Hh    h B   AQ$ !h BMh  AR$ #h BDn@k AT$N2 $h S 8c  8N2 %h S 8c;N2 &h S 8c\ G N2 'h S 8cu` N2 (h S 8cc  \T   |  )h#   | BB *h 3 8c[  |  +h ~BCDEFK @`  \T  4 ,h#  BB -h 3 8c  .h ~BCDEF:: @`x46\T Y /h# BB 0hB 3 8cg 1h ~BCDEFf @`Yr\T A  2h# d BB 3h 3 8cAE  4h ~BCDEF66 @`5&  \T  n  5h#   BB 6h 3 8c1 nB  7h ~BCDEFSyS @`  h \T  @   8h#  @ 2 BB 9h 3 8cS   :h ~BCDEFJ @` @ Y  h 3 8cc 1 ?h ~BCDEFmvm @`^ Ah B E + AS$ Ch 6    KG2: Directed Graph F    Gh     h B AD$  h B<Ժ 0   AB$T2 h c $8c$=T2 h c $8cTT2 h c $8cL v T2 h c $8ceT2 h c $8ct h HB h C 8c  HB h C 8c HB hB C 8cjHB h C 8c HB h C 8c@9 HB h C 8c@ HB h C 8cj  h BV AA$ h <ȃo]U AE$ Eh 0    CC& 2H h 03޽h ? ̙33y___PPT10Y+D=' *= @B +  /'5pl(  l l H     l 0̟p  EThe degree d(v) of a vertex v is the number of edges incident to v. d (A) = three, d (D) = two In directed graphs, indegree is the number of incoming edges at the vertex and outdegree is the number of outgoing edges from the vertex. The indegree of P is 2, its outdegree is 1. The indegree of Q is 1, its outdegree is 1. E 2 $ 3 L   ?l#  sR @l <KR AD$ Al < 0  AB$N2 Bl S 8c$=N2 Cl S 8cTN2 Dl S 8cL v N2 El S 8ceN2 Fl S 8ct h BB Gl 3 8c  BB Hl 3 8c BB IlB 3 8cjBB Jl 3 8c BB Kl 3 8c@9 BB Ll 3 8c@ BB Ml 3 8cj  Nl <U  AA$ Ol 6do  AE$ Pl s *   CC& 2QL    Ql#  a Rl <   AQ$ Sl <i #  AR$ Tl <@ AT$H2 Ul C 8c  8H2 Vl C 8c;H2 Wl C 8c\ G H2 Xl C 8cu` H2 Yl C 8cc  \T   |  Zl#   | BB [l 3 8c[  |  \l ~BCDEFK @`  \T  4 ]l#  BB ^l 3 8c  _l ~BCDEF:: @`x46\T Y `l# BB alB 3 8cg bl ~BCDEFf @`Yr\T A  cl# d BB dl 3 8cAE  el ~BCDEF66 @`5&  \T  n  fl#   BB gl 3 8c1 nB  hl ~BCDEFSyS @`  h \T  @   il#  @ 2 BB jl 3 8cS   kl ~BCDEFJ @` @ Y  ll <D AP$\T c ^ ml# j BB nl 3 8cc 1 ol ~BCDEFmvm @`^ pl <x E T AS$H l 03޽h ? ̙33y___PPT10Y+D=' *= @B +X-  >'6' 7vp&( /_$jw p~ p s *@    p s *@Z 8A path from vertex v1 to vk is a sequence of vertices v1,v2, & , vk that are connected by edges (v1,v2), (v2,v3), & , (vk-1,vk). Path from D to E : (D,A,B,E) Edges in the path : (D,A), (A,B), (B,E) A path is simple if each vertex in it appears only once. DABE is a simple path. ABCDAE is not a simple path. Vertex u is said to be reachable from v if there is a path from v to u. A circuit is a path whose first and last vertices are the same. DAEBCEAD, ABEA, DABECD, SPQRS, STRS are circuits6            Pw *jQL  (j  ?p# %  @p <$ (j  AQ$ Ap <i cU  AR$ Bp <@' AT$H2 Cp C 8c  8H2 Dp C 8c;H2 Ep C 8c\ G H2 Fp C 8cu` H2 Gp C 8cc  \T   |  Hp#   | BB Ip 3 8c[  |  Jp ~BCDEFK @`  \T  4 Kp#  BB Lp 3 8c  Mp ~BCDEF:: @`x46\T Y Np# BB OpB 3 8cg Pp ~BCDEFf @`Yr\T A  Qp# d BB Rp 3 8cAE  Sp ~BCDEF66 @`5&  \T  n  Tp#   BB Up 3 8c1 nB  Vp ~BCDEFSyS @`  h \T  @   Wp#  @ 2 BB Xp 3 8cS   Yp ~BCDEFJ @` @ Y  Zp <D AP$\T c ^ [p# j BB \p 3 8cc 1 ]p ~BCDEFmvm @`^ ^p < ! EN  AS$L   _p#  j `p <%KR AD$ ap <* 0  AB$N2 bp S 8c$=N2 cp S 8cTN2 dp S 8cL v N2 ep S 8ceN2 fp S 8ct h BB gp 3 8c  BB hp 3 8c BB ipB 3 8cjBB jp 3 8c BB kp 3 8c@9 BB lp 3 8c@ BB mp 3 8cj  np <0U  AA$ op 64o  AE$ pp s *8   CC& 2 sp  0e0e    BC0DEFT jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||77\{tpi8`8`f=0  HVf?xXagp(+QCp!00&q`PGA8'(@                  @p,$D0J vp  0e0e    BCDEFX3 jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||::xTzq0Xy3XJ!C` (3 c^xh @cm(S4shQZY P4%),@                   S" `  ,$D0H p 03޽h ? ̙33___PPT10+ηD~' *= @B D9' = @BA?%,( < +O%,( < +D4' (=%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*sp%(D4' (=%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*vp%(+W)  =#5#07t"( t$jw tZ st  0e0e    BCDEF\3 jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||==`AM \PF?3(:P' H)Y:f@xLmxzfJ(fkH8'! 3Y68@RgRnXEt+,@                    S" X ` ,$D0~ t s *[@   7 t s *^&P  3A simple circuit is a cycle if except for the first (and last) vertex, no other vertex appears more than once. ABEA, DABECD, SPQRS, and STRS are cycles. A Hamiltonian cycle of a graph G is a cycle that contains all the vertices of G DABECD is a Hamiltonian cycle of G1 PQRSTP is a Hamiltonian of G2. 5L L   ?t#  4 R @t <KR AD$ At <a 0  AB$N2 Bt S 8c$=N2 Ct S 8cTN2 Dt S 8cL v N2 Et S 8ceN2 Ft S 8ct h BB Gt 3 8c  BB Ht 3 8c BB ItB 3 8cjBB Jt 3 8c BB Kt 3 8c@9 BB Lt 3 8c@ BB Mt 3 8cj  Nt <hU  AA$ Ot 6 o  AE$ Pt s *o   CC& 2QL  (j  Qt#  D ; Rt <Pt (j  AQ$ St <xi cU  AR$ Tt <t|@' AT$H2 Ut C 8c  8H2 Vt C 8c;H2 Wt C 8c\ G H2 Xt C 8cu` H2 Yt C 8cc  \T   |  Zt#   | BB [t 3 8c[  |  \t ~BCDEFK @`  \T  4 ]t#  BB ^t 3 8c  _t ~BCDEF:: @`x46\T Y `t# BB atB 3 8cg bt ~BCDEFf @`Yr\T A  ct# d BB dt 3 8cAE  et ~BCDEF66 @`5&  \T  n  ft#   BB gt 3 8c1 nB  ht ~BCDEFSyS @`  h \T  @   it#  @ 2 BB jt 3 8cS   kt ~BCDEFJ @` @ Y  lt <\D AP$\T c ^ mt# j BB nt 3 8cc 1 ot ~BCDEFmvm @`^ pt <` EN  AS$j tt  0e0e    BCDEF`3 jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||@@@_/l?Xkw8[xQ^OG>70((96(H%2'/ @!079?k3`i2Oc|@-0@                     S"   ,$D0H t 03޽h ? ̙33___PPT10+WD~' *= @B D9' = @BA?%,( < +O%,( < +D4' (=%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*st%(D4' (=%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*tt%(+)  ((@Rx F(( Pԫ$jw x| x s *8&. "A subgraph of a graph G = (V,E) is a graph H(U,F) such that U V and FE. H1 {[U1:A,E,C,D], F1:[ (A,E),(E,C),(C,D),(D,A)]} is a subgraph of G1 H2 {[U2:S,P,T],F2:[(S,P),(S,T),(T,P)]} is a subgraph of G2.f4  x 6 F WSpanning tree of G1(L   Wx# \:~  Xx <\KR AD$ Yx <ȗ 0  AB$N2 Zx S 8c$=N2 [x S 8cTN2 \x S 8cL v N2 ]x S 8ceN2 ^x S 8ct h BB _x 3 8c  BB `x 3 8c BB axB 3 8cjBB bx 3 8c BB cx 3 8c@9 BB dx 3 8c@ BB ex 3 8cj  fx <U  AA$ gx 6o  AE$ hx s *   CC& 2F    {x y *  jx B̼! )  AD$T2 lx c $8c#  T2 mx c $8c# \T2 nx c $8c_ N T2 ox c $8c;HB qx C 8c]  m HB rx C 8c -^HB sxB C 8c;EZHB ux C 8c q}  xx BFH. AA$ yx <| K AE$ zx 0   CC& 2QL  (j  |x# 3  }x < (j  AQ$ ~x <i cU  AR$ x <P@' AT$H2 x C 8c  8H2 x C 8c;H2 x C 8c\ G H2 x C 8cu` H2 x C 8cc  \T   |  x#   | BB x 3 8c[  |  x ~BCDEFK @`  \T  4 x#  BB x 3 8c  x ~BCDEF:: @`x46\T Y x# BB xB 3 8cg x ~BCDEFf @`Yr\T A  x# d BB x 3 8cAE  x ~BCDEF66 @`5&  \T  n  x#   BB x 3 8c1 nB  x ~BCDEFSyS @`  h \T  @   x#  @ 2 BB x 3 8cS   x ~BCDEFJ @` @ Y  x <D AP$\T c ^ x# j BB x 3 8cc 1 x ~BCDEFmvm @`^ x < EN  AS$F   y  x g  x B@t   AT$N2 x S 8c q) N2 x S 8cq d N2 x S 8c V \T  4 x# o  BB x 3 8c  x ~BCDEF:: @`x46\T Y x# B=  BB xB 3 8cg x ~BCDEFf @`Yr x BX2 y  AP$\T c ^ x# S  BB x 3 8cc 1 x ~BCDEFmvm @`^ x B Cn  AS$H x 03޽h ? ̙33y___PPT10Y+D=' *= @B +-  ',,P[` +(  x  c $ \8&}   p  # lO1?  dA spanning tree of a graph G is a subgraph of G that is a tree and contains all the vertices of G.2e V#:L   #  F  <KR AD$  < 0  AB$N2  S 8c$=N2  S 8cTN2   S 8cL v N2   S 8ceN2   S 8ct h BB   3 8c  BB   3 8c BB B 3 8cjBB  3 8c BB  3 8c@9 BB  3 8c@ BB  3 8cj   <U  AA$  6,o  AE$  s *(   CC& 2F |  Z  ( 1 8   B" -   AD$  B& &  AB$T2  c $8c l  T2  c $8c j` T2  c $8c; R T2  c $8cq?  T2  c $8cb Z HB  C 8c9 i q HB  B C 8c? !^ HB ! C 8cDG 3 HB # C 8c y   % B,"L 2  AA$ & <x1 '  AE$ ' 0t5|  CC& 2 L  K[B )#  * 69. C EQ( + 6=Dp & ER( , 6xA H  ET(N2 - S 333] N ` N2 . S 333]cN2 / S 333]% V H2 0 C 333]k V N2 1 S 333] [j \N ]w  2 ]w BB 3 3 ]]6w  4 BCoDEF6o6o @`\N >S  5 >S BB 6 3 ]  7 BCDEF)) @` >S  8 6Gyj  EP(\N 1   9 1  BB : 3 ]1 F  ; BC{DEFU{oU @`R   < 6L K=  ES( = 6hOB WSpanning tree of G2(\N  V  >  V BB ?B 3 ]"   @ BCvDEF>v @` 'V  B BU%  AQ$ C BY] AR$ D B]ut[ AT$F2 E S 8cF2 F S 8cD(F2 G S 8cF2 H S 8c\F2 I S 8cr TL   |  J# BB K 3 8c[  |  L ~BCDEFK @`  TL  4 M# BB N 3 8c  O ~BCDEF:: @`x46TL Y P# 5BB QB 3 8cg R ~BCDEFf @`YrTL A  S#  &BB T 3 8cAE  U ~BCDEF66 @`5&  TL  n  V# BB W 3 8c1 nB  X ~BCDEFSyS @`  h TL  @   Y# BB Z 3 8cS   [ ~BCDEFJ @` @ Y  \ Bg AP$TL c ^ ]# BB ^ 3 8cc 1 _ ~BCDEFmvm @`^ ` BlLX2 AS$H  03޽h ? %9e377y___PPT10Y+D=' *= @B +  `|h(  @ L  |~ | s *    | s *P8^Q PNA graph is said to be connected if there is a path from any vertex to any other vertex in the graph. G1 and G2 are both connected graphs A forest is a graph that does not contain a cycle. A tree is a connected forest. A spanning forest of an undirected graph G is a subgraph of G that is a forest and contains all the vertices of G. If a graph G(V,E) is not connected, then it can be partitioned in a unique way into a set of connected subgraphs called connected components. A connected component of G is a connected subgraph of G such that no other connected subgraph of G contains it. q 2 l-qH | 03޽h ? ̙33y___PPT10Y+D=' *= @B +  p!@m( $jw ~  s * 3 OS<B ? # OS<B @ # OPB A 3 S<B B # SB C 3 <B D # <B E # B F 3 8<B G # 8B H 3 8<<B I # 8<<B J # 89B K 3 <<B L # <B M 3 <B N # <B O # B P 3 ! <B Q # ! B R 3 ! % <B S # ! % <B T # ! " B U 3 %  <B V # %  B W 3   <B X #   <B Y #   B Z 3   <B [ #   <B \ #   B ] 3 <B ^ # B _ 3 OS<B ` # OPB a 3 <B b # B c 3 8<<B d # 89B e 3 <B f # B g 3 ! % <B h # ! " B i 3   <B j #    k 6,  SA60 G l 6 ~ S060 G m 6tU S160 G n 6g S060 G o 6> S160 G p 6P   S160 GB q 3 <B r # <B s # B t 3 O<B u # OB v 3 OS<B w # OS<B x # OPB y 3 S<B z # SB { 3 <B | # <B } # B ~ 3 8<B  # 8B  3 8<<B  # 8<<B  # 89B  3 <<B  # <B  3 <B  # <B  # B  3 ! <B  # ! B  3 ! % <B  # ! % <B  # ! " B  3 %  <B  # %  B  3   <B  #   <B  #   B  3 <B  # B  3 OS<B  # OPB  3 <B  # B  3 8<<B  # 89B  3 <B  # B  3 ! % <B  # ! " B  3   <B  #     6( ` SB60 G  6l-~` S160 G  60U` S060 G  6D4g` S160 G  68>` S060 G  6h<P  ` S160 GB  3 <B  # <B  # B  3 O<B  # OB  3 OS<B  # OS<B  # OPB  3 S<B  # SB  3 <B  # <B  # B  3 8<B  # 8B  3 8<<B  # 8<<B  # 89B  3 <<B  # <B  3 <B  # <B  # B  3 ! <B  # ! B  3 ! % <B  # ! % <B  # ! " B ˆ 3 %  <B È # %  B Ĉ 3   <B ň #   <B ƈ #   B Lj 3 T<B Ȉ # TB Ɉ 3 OST<B ʈ # OPTB ˈ 3 T<B ̈ # TB ͈ 3 8<T<B Έ # 89TB ψ 3 T<B Ј # TB ш 3 ! % T<B ҈ # ! " TB ӈ 3   T<B Ԉ #   T Ո 6DK X+ SC60 G ֈ 6O~X+ S060 G ׈ 6SXU+ S160 G ؈ 6WgX+ S060 G و 6[X>+ S160 G ڈ 6`P X + S160 GB ۈ 3 TX<B ܈ # TU<B ݈ # TXB ވ 3 TOX<B ߈ # TOUB  3 OTSX<B  # OTSU<B  # OTPXB  3 STX<B  # STUB  3 TX<B  # TU<B  # TXB  3 T8X<B  # T8UB  3 8T<X<B  # 8T<U<B  # 8T9XB  3 <TX<B  # <TUB  3 TX<B  # TU<B  # TXB  3 T! X<B  # T! UB  3 ! T% X<B  # ! T% U<B  # ! T" X@@ T  T B  3 % T X<B  # % T UB  3  T X<B  #  T U<B  #  T XB  3 X<B  # XB  3 OXS<B  # OXPB  3 X<B  # XB  3 8X<<B  # 8X9B  3 X<B  # XB  3 ! X% <B  # ! X" B   3  X <B   #  X    6|o # SD60 G   6t~# S160 G   6w#U S060 G  6{g# S160 G  6#> S060 G  6P #  S060 GB  3 #<B  # <B  # #B  3 O#<B  # OB  3 OS#<B  # OS<B  # OP#B  3 S#<B  # SB  3 #<B  # <B  # #B  3 8#<B  # 8B   3 8<#<B ! # 8<<B " # 89#B # 3 <#<B $ # <B % 3 #<B & # <B ' # #B ( 3 ! #<B ) # ! B * 3 ! % #<B + # ! % <B , # ! " #B - 3 %  #<B . # %  B / 3   #<B 0 #   <B 1 #   #B 2 3 #<B 3 # #B 4 3 O#S<B 5 # O#PB 6 3 #<B 7 # #B 8 3 8#<<B 9 # 8#9B : 3 #<B ; # #B < 3 ! #% <B = # ! #" B > 3  # <B ? #  #  @ 6@   SE60 G A 6Ė~  S160 G B 6U  S160 G C 6lg  S160 G D 6>  S060 G E 6ܦP   S060 GB F 3 <B G # <B H # B I 3 O<B J # OB K 3 OS<B L # OS<B M # OPB N 3 S<B O # SB P 3 <B Q # <B R # B S 3 8<B T # 8B U 3 8<<B V # 8<<B W # 89B X 3 <<B Y # <B Z 3 <B [ # <B \ # B ] 3 ! <B ^ # ! B _ 3 ! % <B ` # ! % <B a # ! " B b 3 %  <B c # %  B d 3   <B e #   <B f #   B g 3  <B h #  B i 3   <B j #   <B k #   B l 3   <B m #   <B n #   B o 3  O <B p #  O B q 3 OS <B r # OP B s 3 O S <B t # O S <B u # O P B v 3 S  <B w # S  B x 3  <B y #  B z 3   <B { #   <B | #   B } 3  8 <B ~ #  8 B  3 8< <B  # 89 B  3 8 < <B  # 8 < <B  # 8 9 B  3 <  <B  # <  B  3  <B  #  B  3   <B  #   <B  #   B  3  ! <B  #  ! B  3 ! % <B  # ! " B  3 ! % <B  # ! % <B  # ! " B  3 % <B  # % B  3   <B  #   B  3  <B  #  <B  #  B  3  <B  #  <B  #    64 .   `Adjacency list60 G  6,    SA60 G  6\~   SB60 G  6 r  SD60 G  6g   SE60 GB  3   <B  #   <B  #   B  3   <B  #   <B  #   B  3  O <B  #  O B  3 O S <B  # O S <B  # O P B  3 S  <B  # S  B  3   <B  #   <B  #   B  3  8 <B  #  8 B  3 8 < <B  # 8 < <B  # 8 9 B  3 <  <B  # <  B  3   <B  #   <B  #   B  3   <B  #   <B  #   B  3   <B  #   B  3 O S <B ‰ # O P B É 3   <B ĉ #   B ʼn 3 8 < <B Ɖ # 8 9 B lj 3   <B ȉ #    ɉ 6    SB60 G ʉ 60~   SA60 G ˉ 6, r  SC60 G ̉ 6g   SE60 GB ͉ 3   <B Ή #   <B ω #   B Љ 3  O <B щ #  O B ҉ 3 O S <B Ӊ # O S <B ԉ # O P B Չ 3 S  <B ։ # S  B ׉ 3   <B ؉ #   <B ى #   B ډ 3  8 <B ۉ #  8 B ܉ 3 8 < <B ݉ # 8 < <B މ # 8 9 B ߉ 3 <  <B  # <  B  3   <B  #   <B  #   B  3   <B  #   B  3 O S <B  # O P B  3   <B  #   B  3 8 < <B  # 8 9 B  3   <B  #     6  a  SC60 G  6~ a  SB60 G  6$ ra  SD60 G  68g a  SE60 GB  3   <B  #   <B  #   B  3  O <B  #  O B  3 O S <B  # O S <B  # O P B  3 S  <B  # S  B  3   <B  #   <B  #   B  3  8 <B  #  8 B  3 8 < <B  # 8 < <B  # 8 9 B  3 <  <B  # <  B  3   <B  #   <B  #   B   3  U <B   #  U B   3 O SU <B   # O PU B   3  U <B  #  U B  3 8 <U <B  # 8 9U B  3  U <B  #  U   6 Y , SD60 G  6~Y , SA60 G  6pY r, SC60 G  6@gY , S\60 GB  3 U Y <B  # U V <B  # U Y B  3 U OY <B  # U OV B  3 OU SY <B  # OU SV <B  # OU PY B  3 SU Y <B   # SU V B ! 3 U Y <B " # U V <B # # U Y B $ 3 U 8Y <B % # U 8V B & 3 8U <Y <B ' # 8U <V <B ( # 8U 9Y B ) 3 <U Y <B * # <U V B + 3 U Y <B , # U V <B - # U Y B . 3 Y  <B / # Y  B 0 3 OY S <B 1 # OY P B 2 3 Y  <B 3 # Y  B 4 3 8Y < <B 5 # 8Y 9 B 6 3 Y  <B 7 # Y   8 64$ $ SE60 G 9 6(~$ SA60 G : 6|,$r SB60 G ; 60g$ SC60 GB < 3  $<B = #  !<B > #  $B ? 3  O$<B @ #  O!B A 3 O S$<B B # O S!<B C # O P$B D 3 S $<B E # S !B F 3  $<B G #  !<B H #  $B I 3  8$<B J #  8!B K 3 8 <$<B L # 8 <!<B M # 8 9$B N 3 < $<B O # < !B P 3  $<B Q #  !<B R #  $B S 3 $<B T # $B U 3 <B V # <B W # B X 3 <B Y # <B Z # B [ 3 O<B \ # OB ] 3 O$S<B ^ # O$PB _ 3 OS<B ` # OS<B a # OPB b 3 S<B c # SB d 3 $<B e # $B f 3 <B g # <B h # B i 3 8<B j # 8B k 3 8$<<B l # 8$9B m 3 8<<B n # 8<<B o # 89B p 3 <<B q # <B r 3 $<B s # $B t 3 <B u # <B v # B w 3 <B x # <B y # H  03޽h ? ̙33y___PPT10Y+D=' *= @B +  F>Bh(  ~  s *   QL  G  (# &a  ) < { G  AQ$ * <zi :3  AR$ + <@ AT$H2 , C 8c  8H2 - C 8c;H2 . C 8c\ G H2 / C 8cu` H2 0 C 8cc  \T   |  1#   | BB 2 3 8c[  |  3 ~BCDEFK @`  \T  4 4#  BB 5 3 8c  6 ~BCDEF:: @`x46\T Y 7# BB 8B 3 8cg 9 ~BCDEFf @`Yr\T A  :# d BB ; 3 8cAE  < ~BCDEF66 @`5&  \T  n  =#   BB > 3 8c1 nB  ? ~BCDEFSyS @`  h \T  @   @#  @ 2 BB A 3 8cS   B ~BCDEFJ @` @ Y  C <E AP$\T c ^ D# j BB E 3 8cc 1 F ~BCDEFmvm @`^ G <D D. { AS$f H 0X99?"`H DL F i # F i J 6@;F ^ G2: Directed6 0  G K 6ȡ; WGraph60 G L 6;gu bAdjacency matrix60 G M 6ĩt$4 SP60 G N 6D7t4 SQ60 G O 6lt(4 SR60 G P 6,2t4 SS60 G Q 6P t 4 ST60 GB R 3 p t<B S # p q<B T # p tB U 3 p t<B V # p q<B W # p tB X 3  pt<B Y #  pqB Z 3 pt<B [ # pq<B \ # ptB ] 3 pt<B ^ # pqB _ 3 p t<B ` # p q<B a # ptB b 3  pt<B c #  pqB d 3 pt<B e # pq<B f # ptB g 3 pt<B h # pqB i 3 pt<B j # pq<B k # ptB l 3 p t<B m # p qB n 3  p t<B o #  p q<B p #  p tB q 3  p t<B r #  p qB s 3  p t<B t #  p q<B u #  p tB v 3  p t<B w #  p q<B x #  p tB y 3 t ,<B z # t ,B { 3 t,<B | # t,B } 3 t ,<B ~ # t,B  3 t,<B  # t,B  3 t,<B  # t,B  3  t ,<B  #  t ,B  3  t ,<B  #  t ,  64;0 SP60 G  60 S060 G  670 S160 G  6`0  S060 G  6t20 S060 G  6 0  S060 GB  3 , 0<B  # , -<B  # , 0B  3  ,0<B  #  ,-B  3 ,0<B  # ,-<B  # ,0B  3 ,0<B  # ,-B  3 , 0<B  # , -<B  # ,0B  3  ,0<B  #  ,-B  3 ,0<B  # ,-<B  # ,0B  3 ,0<B  # ,-B  3 ,0<B  # ,-<B  # ,0B  3 , 0<B  # , -B  3  , 0<B  #  , -<B  #  , 0B  3  , 0<B  #  , -B  3  , 0<B  #  , -<B  #  , 0B  3 0 <B  # 0 B  3 0<B  # 0B  3 0 <B  # 0B  3 0<B  # 0B  3 0<B  # 0B  3  0 <B  #  0 B  3  0 <B  #  0   6; SQ60 G  6| S060 G  6x7 S060 G  6$  S160 G  672 S060 G  67   S060 GB Œ 3  <B Ì #  <B Č #  B Ō 3  <B ƌ #  B nj 3 <B Ȍ # <B Ɍ # B ʌ 3 <B ˌ # B ̌ 3  <B ͌ #  <B Ό # B ό 3  <B Ќ #  B ь 3 <B Ҍ # <B ӌ # B Ԍ 3 <B Ռ # B ֌ 3 <B ׌ # <B ، # B ٌ 3  <B ڌ #  B ی 3   <B ܌ #   <B ݌ #   B ތ 3   <B ߌ #   B  3   <B  #   <B  #   B  3  <B  #  B  3 <B  # B  3  <B  # B  3 <B  # B  3 <B  # B  3   <B  #   B  3   <B  #     67;i SR60 G  6D7i S060 G  6(77i S060 G  6@7 i S060 G  6 72i S160 G  6$7  i S060 GB  3  <B  #  <B  #  B  3  <B  #  B  3 <B  # <B  # B  3 <B  # B  3  <B  #  <B  # B  3  <B  #  B  3 <B  # <B  # B   3 <B   # B   3 <B   # <B   # B  3  <B  #  B  3   <B  #   ?L    ۍ#   <B  #   B  3   <B  #   B  3   <B  #   <B  #   B  3  b<B  #  bB  3 b<B  # bB  3  b<B  # bB  3 b<B   # bB ! 3 b<B " # bB # 3   b<B $ #   bB % 3   b<B & #   b ' 637;e%  SS60 G ( 687e%  S160 G ) 6;77e%  S060 G * 6@7e %  S060 G + 6\C72e%  S060 G , 6G7 e %  S160 GB - 3 b e<B . # b c<B / # b eB 0 3  be<B 1 #  bcB 2 3 be<B 3 # bc<B 4 # beB 5 3 be<B 6 # bcB 7 3 b e<B 8 # b c<B 9 # beB : 3  be<B ; #  bcB < 3 be<B = # bc<B > # beB ? 3 be<B @ # bcB A 3 be<B B # bc<B C # beB D 3 b e<B E # b cB F 3  b e<B G #  b c<B H #  b eB I 3  b e<B J #  b cB K 3  b e<B L #  b c<B M #  b eB N 3 e  <B O # e  B P 3 e <B Q # e B R 3 e  <B S # e B T 3 e <B U # e B V 3 e <B W # e B X 3  e  <B Y #  e  B Z 3  e  <B [ #  e   \ 6HV7;"   ST60 G ] 6Z7"   S160 G ^ 6^77"   S060 G _ 6pb7"   S160 G ` 6f72"   S060 G a 6j7 "  S060 GB b 3  " <B c #   <B d #  " B e 3   " <B f #    B g 3  " <B h #   <B i #  " B j 3  " <B k #   B l 3  " <B m #   <B n #  " B o 3   " <B p #    B q 3  " <B r #   <B s #  " B t 3  " <B u #   B v 3  " <B w #   <B x #  " B y 3   " <B z #    B { 3   " <B | #    <B } #   " B ~ 3   " <B  #    B  3    " <B  #     <B  #   " B  3 "  <B  # "  B  3   <B  #   <B  #   B  3   <B  #   <B  #   B  3    <B  #    B  3 "  <B  # "  B  3   <B  #   <B  #   B  3   <B  #   B  3 "  <B  # "  B  3   <B  #   <B  #   B  3    <B  #    B  3 "  <B  # "  B  3   <B  #   <B  #   B  3   <B  #   B  3 "  <B  # "  B  3   <B  #   <B  #   B  3   <B  #   B  3  " <B  #  " B  3  <B  #  <B  #  B  3  <B  #  B  3  "  <B  #  " B  3   <B  #   <B  #  B  3   <B  #   <B  #    67; nU  `Adjacency list60 G  6P7;Q   SP60 G  6h7Q 5  SQ60 G  677Q c  S/60 GB  3 O S <B  # O P <B  # O S B  3 O S <B  # O P <B  # O S B  3  O S <B Í #  O P B č 3 O S <B ō # O P <B ƍ # O S B Ǎ 3 O S <B ȍ # O P B ɍ 3 O S <B ʍ # O P <B ˍ # O S B ̍ 3  O S <B ͍ #  O P B ΍ 3 O S <B ύ # O P <B Ѝ # O S B э 3 O S <B ҍ # O P <B Ӎ # O S B ԍ 3 S  <B Ս # S  B ֍ 3 S  <B ׍ # S  B ؍ 3 S  <B ٍ # S  B ڍ 3 S  :B ܍ 3 S   ݍ <7 ;  SQ60 G ލ <7 -  SR60 G ߍ <7 7c  S/60 G@  C    :B  3    :B  3    @  C   :B  3   @  C   :B  3   :B  3   @  C   :B  3   @  C    :B  3    :B  3   @  C   :B  3   @  C   :B  3   :B  3   @  C    :B  3    @  C   :B  3   @  C    :B  3   @  C   :B  3     <L7 ;  SR60 G  < 7 $  SS60 G  < 7 7c  S/60 G@  C    :B  3    :B  3    @  C   :B  3   @  C   :B  3   :B  3   @  C   :B  3   @  C    :B  3    :B   3   @   C   :B   3   @   C   :B   3   :B  3   @  C    :B  3    @  C   :B  3   @  C    :B  3   @  C   :B  3     <7 ;G SS60 G  <7 $G SP60 G  <7 7G ST60 G@  C    :B  3    :B  3    @  C   :B  3   @  C   :B   3   :B ! 3   @ " C   :B # 3   @ $ C    :B % 3    :B & 3   @ ' C   :B ( 3   @ ) C   :B * 3   :B + 3   @ , C   ?:B - 3   ?@ . C  ?:B / 3  ?@ 0 C   ?:B 1 3  ?@ 2 C  ?:B 3 3  ? 4 <7C; ST60 G 5 <`7C$ SP60 G 6 <`7C7 SR60 G@ 7 C ? C:B 8 3 ? @:B 9 3 ? C@ : C ? C:B ; 3 ? @@ < C ?C:B = 3 ?@:B > 3 ?C@ ? C ?C:B @ 3 ?@@ A C ? C:B B 3 ? @:B C 3 ?C@ D C ? C:B E 3 ? @@ F C ?C:B G 3 ?@:B H 3 ?C@ I C C :B J 3 C @ K C  :B L 3  :B M 3  @ N C  :B O 3  :B P 3  @ Q C  :B R 3  @ S C C:B T 3 C@ U C :B V 3 :B W 3 @ X C :B Y 3 @ Z C C :B [ 3 C@ \ C  :B ] 3  :B ^ 3 @ _ C  :B ` 3  @ a C C:B b 3 C@ c C :B d 3 :B e 3 @ f C :B g 3 :B h 3 H  03޽h ? ̙33y___PPT10Y+D=' *= @B +;   bZ(  ~  s *ܼ     s *w NLProcedure DFS_Tree G(V,E) Input: G = (V,E); S is a stack - initially empty;  x refers to the top of stack; initially mark all vertices  new ; L[x] refers to the adjacency list of x. T {0}; Output : The DFS tree T; 1. v old; v V 2. push (S,v); 3. while S is nonempty do 4. while there exists a vertex w in L[x] and marked new do 5. T T (x,w) ; 6. w old; 7. push w onto S 8. pop S  2  2)  3 rw1?yf  O (V + E )(2$$$$$$$$$H  03޽h ? ̙33y___PPT10Y+D=' *= @B +     >V (  ~  s *    ' 6DG4 EE( ) 6L}_ EA( + 6J=HS  EC( - 6tS_#u ED( / 6Vb-  EB(@2 0 C 3338@2 1 C 333i@2 2 C 333gM@2 3 C 333ku@2 4 C 333{Z4B 5 # 4B 6 # W^4B 7@ # ;;4B 8 # e4B 9 # W#4B : # #4B ; # ;F2 P S jJO[@ Q C p Ow S <^ss P EA( U <cck@ EB(F2 V S jJ    c $aop  > F2  S 8c?   BCDEF333 @`~9 ?   <j|   CA&   <oY CB&F2   S 8c@ F2  S 8cV  BcCDEF cc @`t  <s CC&  ByC"DEF"yy @`x ? = 6(x 8  EE( ? 6P| {   EA( A 6 C  EB( C 6 w Y  EC(F2 D S o w  E BCDEF333 @`F P F2 F S o J F2 G S o   H BCDEF @`@ z M  I BCDEF  @`B  P F2 J S o w  K BkCDEFck  @` q  M 6%w }  ED(F2 N S oJ a#  O BcCDEFcO{c @`  G   < +w l  CA&  <ls f  CB&  <`1w J  CC&F2  S 8c    BCDEF333 @`F P F2  S 8c I F2  S 8c @   BqCDEFqq @`@ FM    BCDEF  @`B QP F2 ! S 8c   "  B;CDEFo5;  @`    $ <7w  CD&dB % <DjJ8 G@ H  03޽h ? ̙33y___PPT10Y+D=' *= @B +V  e],(  ~  s *\w  w   s *hws,$ 0 rInitially, T = {0}; S {0}, A,B,C,D,E (all new) Starts at A : A, S : {A}, L[A] = {B,D,E} Pick B from L[A]; T = {(A,B)} and B (it's marked old} S = {A,B}, L[B] = {A,C,E} Pick C from L[B]; T = {(A,B), (B,C)} and C S = {A,B,C}; L[C] = {B,D,E} Pick D from L[C] ; T = {(A,B), (B,C), (C,D)} and D S = { A,B,C, D} ; L[D] ={A,C}; no new vertices; S = { A,B,C}; L[C] = { B,D,E} Pick E from L[C]; T ={ (A,B), (B,C), (C,D),(C,E)} and E S = { A,B,C,E} ; L[E] = {A,B,C} S = { A,B,C}; L[C] = { B,D,E} S ={ A,B} ; L[B]= { A,C,E } S ={A} ; L[A] = { B,C,E} S = {0} pr 2>7  !       L  n  #    <@jw3 AD$  <w 0 n  AB$N2  S 8c$=N2  S 8cTN2   S 8cL v N2 ! S 8ceN2 " S 8ct h BB # 3 8c  BB $ 3 8c BB %B 3 8cjBB & 3 8c BB ' 3 8c@9 BB ( 3 8c@ BB ) 3 8cj  * <wU/ AA$ + 6woy AE$ , s *w g  CC& 2H  03޽h ? ̙33/E'E___PPT10E+DD' *= @B DnD' = @BA?%,( < +O%,( < +D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*/D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*/D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*/[%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*/[D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*/[D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*[%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*[D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*[D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*N%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*ND' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*ND{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*N%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*ND' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*ND{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*B%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*BD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*BD{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*B\%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*B\D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*B\D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\q%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*\qD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*\q+8+0+  +  aa @  `(   x  c $8&}   F a8J  z ]   B\ AD$  BQ J  AB$T2  c $8cusT2  c $8cj8T2  c $8c<bT2   c $8cCgT2   c $8cwHB   C 8c;^rHB   C 8cr`7HB  B C 8c%s)HB  C 8c}HB  C 8ctGHB  C 8cHB  C 8c%)  B   AA$  <̯P AE$  0\a  CC& 2z (  (,$D 0 T p  x #  (F@ p  x p  xB  3 rDjJ?p p xB  3 rDjJ?  xB  3 rDjJ?p x xB  3 rD1?p    6c  r AA$T2  c $8c Y{  B  AA$z X 0  X 0,$D 0T2 ) c $8cH * BdnJ AA$T2 + c $8cXuB ,B 3 rDjJ? - BŞh AB$@ XX 0 XX 0ZT p  x ## XX 0B $ 3 rDjJ?p p xB % 3 rDjJ?  xB & 3 rDjJ?p x xB ' C xD1?Y8 8 ( <DʞK<" AA$B . C xD1?X0 0 / <HPG- AB$-z    a  x,$D 0ZT p  x 2#    B 3 3 rDjJ?p p xB 4 3 rDjJ?  xB 5 3 rDjJ?p x xB 6 C xD1?   7 <Ӟ  T  EA(B = C xD1?    > <,؞   _  EB( C 0ܞ    GC* 2B D C xD1?    z )  0    ) 0 ,$D 0T2 8 c $8c % * | 9 B ;   AA$T2 : c $8c 5 = B ;B 3 rDjJ? h h 6 < B @   AB$B @ C xD1? p p XT2 A c $8c - b  B 0 )  0  CC& 2T    o#   ` 8ZT p  x p#   B q 3 rDjJ?p p xB r 3 rDjJ?  xB s 3 rDjJ?p x xB t 3 rD1?  u 6՞ T  EA(B v 3 rD1?   w 6|  _  EB( x s *   GC* 2B y 3 rD1?  z !   ! ,$D 0T W  ]# !T2 M c $8c?  N <e AA$T2 O c $8cOB PB 3 rDjJ? Q <_ AB$B T C xD1?0T2 U c $8cGD V 0 CC& 2B Z 3 rD1?H( N2 [ S 8c2 W  \ 6   AD$T 8  c#  B d 3 rDjJ?88 B e 3 rDjJ? B f 3 rDjJ?8  B g 3 rD1?9   h 6 ' \  EA(B i 3 rD1?8 j 6)g  EB( k s * GC* 2B l 3 rD1?8 m <H* EE(B n 3 rD1?8T2 | c $8cXYB } C xD1?pP ` ~ BT: AE$    #  9,$D 0ZT p  x #   B  3 rDjJ?p p xB  3 rDjJ?  xB  3 rDjJ?p x xB  3 rD1?   6d$ T  EA(B  3 rD1?    6(  _  EB(  s *,   GC* 2B  3 rD1?   XX 0 #  ',$D 0ZT p  x # XX 0B  3 rDjJ?p p xB  3 rDjJ?  xB  3 rDjJ?p x xB  3 rD1?Y8 8  60K<" AA$B  3 rD1?X0 0  65PG- AB$E p  x #  jp,$D  0TN p  x  p  xB  3 rDjJ?p p xB  3 rDjJ?  xB  3 rDjJ?p x xB  3 rD1?p    6|:c  r AA$ p  x #  h,$D  0B  3 rDjJ?p p xB  3 rDjJ?  xB  3 rDjJ?p x xRL p  x # `C B  3 rDjJ?p p xB  3 rDjJ?  xB  3 rDjJ?p x x(z `<yB  <`yB,$D 0@ 8  b yB H C xDjJ? 88 B I C xDjJ?  B J C xDjJ? 8  B K C xD1? 9   L <dA ' \  EA(B R C xD1? 8 S <E )g  EB( W 0I  GC* 2B X C xD1? 8 ^ BM * ED(B _ C xD1? 8~T W  # `<BN2  S 8c?   6Re AA$N2  S 8cOB B # lDjJ?  6V_ AB$B  3 rD1?0N2  S 8cGD  s *[ CC& 2B  3 rD1?H( N2  S 8c2 W   6P_  AD$H  03޽h ? %9e37755___PPT105+FDr5' *= @B D-5' = @BA?%,( < +O%,( < +D6' (=%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D6' (=%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%( /%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D6' (=%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D6' (=%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D6' (=%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%(/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*a%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*aD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*aD6' (=%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%(#/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D6' (=%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%()/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D6' (=%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%(//%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D6' (=%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%(5/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D6' (=%(D' =%(D' =4@BBBB%(E' =4 B`BPB`B?<*%(;/%,( < +D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*+   ,$(  ~  s *DWw8&}  w   s *8w@Vz Procedure BFS_Tree G(V,E) Input: G = (V,E); Q is a queue - initially empty; x Q : remove the front item of queue and denote it by x; initially mark all vertices  new ; L[x] refers to the adjacency list of x. T {0} Output: The BFS tree T; 1. v old; v V 2. insert (Q,v); 3. while Q is nonempty do 4. x Q 5. for each vertex w in L[x] and marked  new 6. T T {x,w} ; 7. w old; 8. insert (Q,w); 0 1  3 r4w1? #  ZBreadth-first search*(2 H  03޽h ? ̙33y___PPT10Y+D=' *= @B +$  KC=U@(  ~  s *w  w   6wq G t CA&  6wy  CB&JL b ' # T r'BB  3 li N2  S lib  'N2   S li' " 6w=<@ CE& % <w h  CA& ' <lw- CB&JL  > (# y BB ) 3 jJ ./N2 * S jJ ! N2 + S jJ>:B , 3 jJ F2 - S jJB6 2 Bw T CC& 4 <(Cw m  CD& 5 <HFwO R  CA& 6 <`Mw s  CB&@B 7 C jJ  B L2 8 c $jJu D  L2 9 c $jJu  @B : C li D  L2 ; c $li [Q  = <w* - CE&@B > C li  L2 ? c $li  @B @ C jJ jN L2 A c $li :  D <wm CD& G <w  CA& I <wR CB&JL  _K J#  KBB K 3 jJ N2 L S jJ @ KN2 M S jJ_K@B N C li<E XL2 O c $li[ Q 6wn q  CE&:B R 3 li< d F2 S S li)  F 1h<  U h1<   6{shk CA&  6H{1   CC&  6{Ups CD&   6 {.  CB&N2   S jJ38N2   S jJXUN2   S jJ/J N2  S jJ2N2  S jJ < g BB  3 jJBB  3 jJ BB B 3 jJXBB  3 jJ E BB  3 jJ& BB  3 jJ BB  3 jJX  T 6{ CE&H  03޽h ? ̙33y___PPT10Y+D=' *= @B +aQ  ]U,(  ~  s *w  w   s *>{lV,$ 0 LInitially, T = {0}; Q {0}, A,B,C,D,E (all new) Starts at A : A, Q : {A}, L[A] = {B,D,E} Pick B from L[A]; T = {(A,B)} and B (it's marked old} Q = {B}, L[A] = {B,D,E} Pick D from L[A]; T = {(A,B), (A,D)} and D Q = {B,D}; L[A] = {B,D,E} Pick E from L[A] ; T = {(A,B), (A,D), (A,E)} and E Q = { B,D,E} ; L[A] ={B,D,E}; no new vertices; Dequeue, Q = {D,E} L[B] = { A,C,E}; Pick C from L[B]; T ={ (A,B), (A,D), (A,E),(B,C)} and C Q = {E, C} ; L[D] = {A,C} Q = {C} ; L[E] = {A,B,C} Q = { 0) ; L[C] = (B,C,E) Q = {0}; L 2>8  "      &   L 1h< P  # ! 9  6P@{sh+ CA&  6`f{1 P  CC&  6i{Vp  CD&  68m{.   CB&N2   S jJ38N2 ! S jJXUN2 " S jJ/J N2 # S jJ2N2 $ S jJ < g BB % 3 jJBB & 3 jJ BB 'B 3 jJXBB ( 3 jJ E BB ) 3 jJ& BB * 3 jJ BB + 3 jJX  , 6s{7 CE&H  03޽h ? ̙33@@___PPT10@+D0@' *= @B D?' = @BA?%,( < +O%,( < +D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*/D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*/D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*/[%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*/[D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*/[D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*[%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*[D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*[D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*N%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*ND' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*ND{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*N%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*ND' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*ND{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4D{' (=%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4K%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*4KD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*4K+8+0+  +? P/?s(    H˟ Z CC&  Bϟ s  CD&  B͟O X  CA&  BHן y  CB&@B   C li  B L2   c $333liu D  L2   c $333liu  @B   C li D  L2   c $333li [Q   B۟* 3 CE&@B  C li  L2  c $333li  @B  C li jN L2  c $333li :   <; EE(  Bo ~ x EA(  Be@n EB(  B4?\H EC(F2  S oaw  BCDEF333 @` F2   S ofJ F2 ! S oY " BCDEF @`z  # BCDEF  @` F2 $ S ow- % BkCDEFck  @`q ' <azj ED(F2 ( S oa#h ) BcCDEFcO{c @`  + <[G CE& - <] CA& / <H4{  CC& 1 < CD& 3 <q.]  CB&F2 4 S 333MoF2 5 S 333KF2 6 S 3330F2 7 S 333+F2 8 S 333%?:B 9 3 :B : 3 l:B ;@ 3 m:B < 3 i.:B = 3 :B > 3 R:B ? 3 GH  03޽h ? ̙33   H(  ܭ ~  s *&{  {   s *7 ,The connected component of a graph G = (V,E) is a maximal set of vertices U V such that for every pair of vertices u and v in U, we have both u and v reachable from each other. In the following we give an algorithm for finding the connected components of an undirected graph.JL  s *{PF <\Procedure Connected_Components G(V,E) Input : G (V,E) Output : Number of Connected Components and G1, G2 etc, the connected components 1. V' V; 2. c 0; 3. while V' 0 do 4. choose u V' ; 5. T all nodes reachable from u (by DFS_Tree) 6. V' V' - T; 7. c c+1; 8. Gc T; 9. T 0; / # P  0   H  03޽h ? ̙33y___PPT10Y+D=' *= @B +[  z(  P   0%{8 : 2 8  0ț{Z  BSuppose the DFS tree starts at A, we traverse from A B C D and do not explore the vertices F, G, and H at all! The DFS_tree algorithm does not work with graphs having two or more connected parts. We have to modify the DFS_Tree algorithm to find a DFS forest of the given graph. z! 26H  03޽h ? ̙33y___PPT10Y+D=' *= @B +w   .(  {  ~  s *T"{  { p  s *{ Procedure DFSForest _G(V,E) Input: G = (V,E); S is a stack - initially empty;  x refers to the top of stack; initially mark all vertices  new ; L[x] refers to the adjacency list of x. F {0}; The DFS Forest Output: The DFS tree F; 1. For each vertex v V do 2. if v is new 3. v old; 4. push (S,v); 5. while S is nonempty do 6. while there exists a vertex w in L[x] and marked new do 7. F F (x,w) ; 8. w old; 9. push w onto S 10. pop S " 2   :)H  03޽h ? ̙33y___PPT10Y+D=' *= @B +;"  b!Z!0HW (   W # lf1? ~  s * {8&}  {   0({@&` : 2   6 { Yn 5H  6{ L n 5G   6|{ ;| 5F   6{    5D  6`{ Yr  5C  6{   5B  6D{ Xr  5AF rjO  V rjO   6{^i O  5D  6{Q @ &  5C  6T{ Q h 5B  6{r 5AN2  S 8cN2  S 8c ^ N2  S 8c  N2  S 8c* p H@ t  !t BB  3 8cte z   ~BCDEFB @`  H@   p  $  p BB " 3 8c ^  # ~BCDEF h @`  p H@ z -  'z - BB %B 3 8cz  & ~BCDEFC @`- H@ (C-  *(C- BB ( 3 8c|HC-  ) ~BCDEF. @`(\H@ W K  -W K BB + 3 8c< K  , ~BCDEFB @`W G  / 6{ M 3  5H 1 6{6j  5G 3 6T{G 5FH2 4 C 8cC z *H2 5 C 8c  = N2 6 S 8c=H@   d 9  dBB 7 3 8c  8 ~BCDEFmum @`K dH@ u t  <u t BB :B 3 8c  ; ~BCDEFN @`u t% H@ 1 q  ?1 q BB =B 3 8c q - > ~BCDEF3 @`1  @2 @ C 8cC : @2 A C 8c* % NF 0  D 0  BB B 3 8cz {  C ~BCDEFJ @` 0 F2 E S 8c* | , F2 F S 8c7 3 NF 7 > C  I > 7 C BB G 3 8c7  H ~BCDEFI @` > C NF  >   L >  BB J 3 8c   K ~BCDEFI @`>  @2 M C 8c# 2 NF *  P *  BB N 3 8ct u  O ~BCDEFJ @` * F2 Q S 8c# = @2 R C 8c1 C NF G 8 S  U 8 G S BB S 3 8cG  T ~BCDEFI @` 8 S H  03޽h ? ̙33y___PPT10Y+D=' *= @B +  @H(  ~  s *{@  {    6.{@Z  { H  03޽h ? ̙33y___PPT10Y+D=' *= @B +}  P$(  r  S }8&}  } r  S }8& } H  03޽h ? 33___PPT10i.v#+D=' *= @B +}  `$(  r  S 0(}8&}  } r  S (}8& } H  03޽h ? 33___PPT10i.v0|C+D=' *= @B +o  ~p (  r  S  }8&}  }  0 TA ? 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?^"_8qy"ljv5Y/f\]R/J_V+5BP@`M*!ThF[uw߁=> ?? 7gE=y|_G8vdmKJ 5l$WRI%y_q[ Kv;`~vI _"ȋ7 a^Ÿ~Bęxň[#d#/_f辚tT|yQOubYH^ ,+U_ :mm6]}?q~~8E#{!0u]j&yc;<Oكy=x`41 y2?v9υ LRџiπ^aWB)B-|C; "T3 0u]j\"D"2 .\(ùNq#SI< zПL^ J/7Co++`W JJ يxw!R 0fxjv5sJny|ؐF8betOGy ER=B26[Ω3Q9=.bm7P+M󳈶 Jcӯ<tXi,|׼6qo,=7WXS ;?l֐V4i)AR$-%IUH hD% S TZA/Z"5Qϙ3wq29|ց|XM)xutm`E~;l"o|-6i^@VBZV#((>Z;SyxӕGjORs9r0uI?luC77A@OP ^8^ViX< M £~298pQ,{sz L(npu#?kǰF!8%cq짍8oX{ OA,qá&gfiH\z.s!#ab3sh7)&fyl?u~P[JϷt0ko"vM w<^7L>gk:{_Vcy5XO3:exjXΪ!Fm% ^wc3w9.7aZ,cxHpK]T/PP5c/΢QIX3M9Z9Kz=$g|"%[D7 {NZTAL $Uh|K\UwvLU]2&pVWVU Ucҩ:S厪l5 Ψ*2fv`!V輟!]`\xڕO@ߝNBLi"*Q@] B$c%[")+!]C C`fPH=.b}~0,ΘvdAT~)2+|O` u$F`C铂rsq7)Ơ\VѯUVW{/$ʷ^ F.8 >w'E_~\ Z J6o!In1w\ܖ y/"Rf7rs[X|[CqgT^dWe=SVdGyڴd*70KOS~?'b&);8K8Q>qW\.}?M白^%Sܥ޽u"@*HQ߃2$vu!<|6 2$p0=a 2$fXL.0hg2$nRg`\I 2$`tzO:;@  '2$T."+;n/]12$V輟!]6c $@_uʚ;2Nʚ;g4CdCdv 0 ppp@ <4!d!d` 0Lƒ<4dddd` 0Lƒ<4BdBd` 0Lƒ)Week 4<Single Source Shortest Paths All Pairs Shortest Path ProblemSingle-Source Shortest Paths 0b$` Single Source Shortest Path$(In a shortest-paths problem, we are given a weighted, directed graph G = (V,E), with weights assigned to each edge in the graph. The weight of the path p = (v0, v1, v2, & , vk) is the sum of the weights of its constituent edges: v0 v1 v2 . . . vk-1 vk The shortest-path from u to v is given by d(u,v) = min {weight (p) : if there are one or more paths from u to v = otherwiseZ :W )The single-source shortest paths problem  *(  Dijkstra's Algorithm : At every step of the algorithm, we compute, d[y] = min {d[y], d[x] + w(x,y)}, where x,y V. Dijkstra's algorithm is based on the greedy principle because at every step we pick the path of least path. C4mF  o$ m c&Dijkstra's Single-source shortest path''  hProcedure Dijkstra's Single-source shortest path_G(V,E,u) Input: G =(V,E), the weighted directed graph and v the source vertex Output: for each vertex, v, d[v] is the length of the shortest path from u to v. mark vertex u; d[u] 0; for each unmarked vertex v V do if edge (u,v) exists d [v] weight (u,v); else d[v] ; while there exists an unmarked vertex do let v be an unmarked vertex such that d[v] is minimal; mark vertex v; for all edges (v,x) such that x is unmarked do if d[x] > d[v] + weight[v,x] then d[x] d[v] + weight[v,x]:5P /P>  !     !  Complexity of Dijkstra's algorithm: Steps 1 and 2 take Q (1) time Steps 3 to 5 take O(V) time The vertices are arranged in a heap in order of their paths from u Updating the length of a path takes O(log V) time. There are V iterations, and at most E updates Therefore the algorithm takes O((E+V) log V) time.lBZ7H$* *  All-Pairs Shortest Path Problem 4!b b       %Floyd' s Algorithm for Shortest Paths&& %  Procedure FLOYDs_G=[V,E] Input: nn matrix W representing the edge weights of an n-vertex directed graph. That is W =w(i,j) where, (Negative weights are allowed) Output: shortest path matrix, dist(i,j) is the shortest path between vertices i and j. for v 1 to n do for w 1 to n do dist[v,w] arc[v,w]; for u 1 to n do for v 1 to n do for w 1 to n do if dist[v,u] + dist[u,w] < dist[v,w] then dist[v,w] dist[v,u] + dist[u,w] Complexity : Q(n3)(P U( '       .      Transitive Closure  Given a directed graph G=(V,E), the transitive closure C =(V,F) of G is a directed graph such that there is an edge (v,w) in C if and only if there is a directed path from v to w in G. Security Problem: the vertices correspond to the users and the edges correspond to permissions. The transitive closure identifies for each user all other users with permission (either directly or indirectly) to use his or her account. There are many more applications of transitive closure. The recursive definition for transitive closure is , Y  ,Warshall's Algorithm for Transitive Closure  -+ ("  " ZProcedure WARSHALL's(G=[V,E]) Input: nn matrix A representing the edge weights of an n-vertex directed graph. That is a =a(i,j) where, Output: transitive closure matrix, t(i,j) =1 if there is a path from i to j, 0 otherwise for v 1 to n do for w 1 to n do t[v,w] a(v,w) for u 1 to n do for v 1 to n do for w 1 to n do if NOT t[v,w] then t[v,w] t[v,u] AND t[u,w] return T&P b Q  .     ` ` ̙33` 333MMM` ff3333f` f` f` 3>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> f(    6t P  T Click to edit Master title style! !  0   RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0 ``  >*  0` `   @*  0 `   @*H  0޽h ? ̙33 Default Design<  0 <(   ~  s * P   ~  s *x#  H  0޽h ? 3380___PPT10.Vf  @V(  ~  s *!P    0 TA ? "4 39     # l\1?@  FA motorist wishes to find the shortest possible route from from Perth to Brisbane. Given the map of Australia on which the distance between each pair of cities is marked, how can we determine the shortest route? 80 2g H  0޽h ? 33___PPT10i.v0+D=' *= @B +  P<(  ~  s *,NP   ~  s *l  H  0޽h ? 33___PPT10i.v[+D=' *= @B +  "`(  ~  s *4     0@s VGiven G (V,E), find the shortest path from a given vertex u V to every vertex v V ( u v). For each vertex v V in the weighted directed graph, d[v] represents the distance from u to v. Initially, d[v] = 0 when u = v. d[v] = if (u,v) is not an edge d[v] = weight of edge (u,v) if (u,v) exists. Dijkstra's Algorithm : At every step of the algorithm, we compute, d[y] = min {d[y], d[x] + w(x,y)}, where x,y V. 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s 2~v0___PPT106___PPT9 j116 " " # @` !$ 3 r1? 2~v0___PPT106___PPT9 j126 " " # @` "$ 3 r1?2~v0___PPT106___PPT9 i86 " " # @` #$ 3 rD1?2~v0___PPT106___PPT9 i76 " " # @` $$ 3 r\1?2~v0___PPT106___PPT9 i36 " " # @` %$ 3 r`1?2~v0___PPT106___PPT9 i56 " " # @` &$ 3 r1?)2~v0___PPT106___PPT9 i16 " " # @` '$ 3 r1?C)2~v0___PPT106___PPT9 i06 " " # @` ($ 3 rǫ1?'C2~v0___PPT106___PPT9 ih6 " " # @` )$ 3 rЫ1?'2~v0___PPT106___PPT9 i66 " " # @` *$ 3 r ګ1?D  ~v0___PPT106___PPT9 mf,g,h6 " " # @` +$ 3 rH1?s D ~v0___PPT106___PPT9 i96 " " # @` ,$ 3 r1? s ~v0___PPT106___PPT9 j116 " " # @` -$ 3 r1? ~v0___PPT106___PPT9 j126 " " # @` .$ 3 rL1?~v0___PPT106___PPT9 i86 " " # @` /$ 3 r$ 1?~v0___PPT106___PPT9 i76 " " # @` 0$ 3 r\1?~v0___PPT106___PPT9 i36 " " # @` 1$ 3 rh1?~v0___PPT106___PPT9 i56 " " # @` 2$ 3 rt!1?)~v0___PPT106___PPT9 i16 " " # @` 3$ 3 r(%1?C)~v0___PPT106___PPT9 i06 " " # @` 4$ 3 r71?'C~v0___PPT106___PPT9 ie6 " " # @` 5$ 3 r@1?'~v0___PPT106___PPT9 i56 " " # @` 6$ 3 rI1?D + ~v0___PPT106___PPT9 oe,f,g,h6 " " # @`V 7$ 3 rR1?s +D F>6___PPT9 ^* "   @` 8$ 3 r_1? +s ~v0___PPT106___PPT9 j116 " " # @` 9$ 3 rdd1?+ ~v0___PPT106___PPT9 j126 " " # @` :$ 3 rr1?+~v0___PPT106___PPT9 i86 " " # @` ;$ 3 rm1?+~v0___PPT106___PPT9 i76 " " # @` <$ 3 r1?+~v0___PPT106___PPT9 i36 " " # @` =$ 3 r<1?+~v0___PPT106___PPT9 i56 " " # @` >$ 3 r1?)+~v0___PPT106___PPT9 i16 " " # @` ?$ 3 r1?C+)~v0___PPT106___PPT9 i06 " " # @` @$ 3 r1?'+C~v0___PPT106___PPT9 id6 " " # @` A$ 3 rЭ1?+'~v0___PPT106___PPT9 i46 " " # @` B$ 3 r,1?D ~ +~v0___PPT106___PPT9 q d,e,f,g,h6 "  " # @`V C$ 3 rؿ1?s ~D +F>6___PPT9 ^* "   @`V D$ 3 rǬ1? ~s +F>6___PPT9 ^* "   @` E$ 3 rԬ1?~ +~v0___PPT106___PPT9 j126 " " # @` F$ 3 rL٬1?~+~v0___PPT106___PPT9 i86 " " # @` G$ 3 r1?~+~v0___PPT106___PPT9 i76 " " # @` H$ 3 r1?~+~v0___PPT106___PPT9 i36 " " # @` I$ 3 r1?~+~v0___PPT106___PPT9 i56 " " # @` J$ 3 rl1?)~+~v0___PPT106___PPT9 i16 " " # @` K$ 3 rD 1?C~)+~v0___PPT106___PPT9 i06 " " # @` L$ 3 rL1?'~C+~v0___PPT106___PPT9 ib6 " " # @` M$ 3 rH 1?~'+~v0___PPT106___PPT9 i36 " " # @` N$ 3 r1?D  ~~v0___PPT106___PPT9 s b,d,e,f,g,h6 "  " # @`V O$ 3 r-1?s D ~F>6___PPT9 ^* "   @`V P$ 3 rh$1? s ~F>6___PPT9 ^* "   @` Q$ 3 r|B1? ~~v0___PPT106___PPT9 j126 " " # @`V R$ 3 rE1?~F>6___PPT9 ^* "   @` S$ 3 rS1?~~v0___PPT106___PPT9 i76 " " # @` T$ 3 rW1?~~v0___PPT106___PPT9 i36 " " # @` U$ 3 r]1?~~v0___PPT106___PPT9 i56 " " # @` V$ 3 ri1?)~~v0___PPT106___PPT9 i16 " " # @` W$ 3 rx1?C)~~v0___PPT106___PPT9 i06 " " # @` X$ 3 rs1?'C~~v0___PPT106___PPT9 ic6 " " # @` Y$ 3 r1?'~~v0___PPT106___PPT9 i26 " " # @` Z$ 3 r1?D $ ~v0___PPT106___PPT9 u b,c,d,e,f,g,h6 "  " # @`V [$ 3 rh1?s $D F>6___PPT9 ^* "   @`V \$ 3 rȟ1? $s F>6___PPT9 ^* "   @`V ]$ 3 rԧ1?$ F>6___PPT9 ^* "   @`V ^$ 3 r1?$F>6___PPT9 ^* "   @` _$ 3 rT1?$~v0___PPT106___PPT9 i96 " " # @` `$ 3 r1?$~v0___PPT106___PPT9 i36 " " # @` a$ 3 r,Ϯ1?$~v0___PPT106___PPT9 i56 " " # @` b$ 3 r8ʮ1?)$~v0___PPT106___PPT9 i16 " " # @` c$ 3 rtܮ1?C$)~v0___PPT106___PPT9 i06 " " # @` d$ 3 r1?'$C~v0___PPT106___PPT9 ia6 " " # @` e$ 3 r1?$'~v0___PPT106___PPT9 i16 " " # @` f$ 3 r1?D  $~v0___PPT106___PPT9 wa,b,c,d,e,f,g,h6 " " # @`V g$ 3 r@1?s D $F>6___PPT9 ^* "   @`V h$ 3 r<1? s $F>6___PPT9 ^* "   @`V i$ 3 r1? $F>6___PPT9 ^* "   @`V j$ 3 r1?$F>6___PPT9 ^* "   @` k$ 3 r|&1?$~v0___PPT106___PPT9 i96 " " # @`V l$ 3 r*1?$F>6___PPT9 ^* "   @` m$ 3 r71?$~v0___PPT106___PPT9 i56 " " # @` n$ 3 r;1?)$~v0___PPT106___PPT9 i16 " " # @` o$ 3 rA1?C)$~v0___PPT106___PPT9 i06 " " # @` p$ 3 rN1?'C$~v0___PPT106___PPT9 iu6 " " # @` q$ 3 r\1?'$~v0___PPT106___PPT9 i06 " " # @` r$ 3 rW1?s D ~v0___PPT106___PPT9 ih6 " " # @` s$ 3 r,o1? s ~v0___PPT106___PPT9 ig6 " " # @` t$ 3 r8j1? ~v0___PPT106___PPT9 if6 " " # @` u$ 3 rt|1?~v0___PPT106___PPT9 ie6 " " # @` v$ 3 rЊ1?~v0___PPT106___PPT9 id6 " " # @` w$ 3 r1?~v0___PPT106___PPT9 ic6 " " # @` x$ 3 r1?~v0___PPT106___PPT9 ib6 " " # @` y$ 3 r(1?)~v0___PPT106___PPT9 ia6 " " # @` z$ 3 rd1?C)~v0___PPT106___PPT9 iu6 " " # @` {$ 3 r1?D ~v0___PPT106___PPT9 yUnmarked vertices6 " " # @` |$ 3 rp1?CD ~v0___PPT106___PPT9 |Distance to vertex8( " " # @` }$ 3 r¯1?'CD___PPT10$f___PPT9H@ Vertex to be markedb "    " # "    @`  ~$ 3 rѯ1?'D___PPT10$b___PPT9D<  Step #b "  " # "    @`B $  `1 ? B $  `1 ? B $  `1 ?B $  `1 ? B $  `1 ? B $  `1 ?''B $  `1 ?CCB $  `1 ?))B $  `1 ?CD B $  `1 ?D D B $  `1 ?B $  `1 ?B $  `1 ?B $  `1 ?B $  `1 ?B $  `1 ? B $  `1 ?s s B $  `1 ?$ $B $  `1 ? B $  `1 ?~ ~B $  `1 ?+ +B $  `1 ? B $  `1 ? B $  `1 ?2 2B $  `1 ?  $ S ~D1??? :  H $ 0޽h ? 33___PPT10i.`+D=' *= @B +  8(  8b 8 C A ?"cA  H 8 0޽h ? 33___PPT10i.+D=' *= @B +t )s!sWW<r(  < < S ~1?oo :  7p pP q  <3 #"2&  < 3 rа1? P q ~v0___PPT106___PPT9 {04 " " #   @`j < 3 r Ӱ1? q F>6___PPT9 r* "     @`j < 3 r@1?R q F>6___PPT9 r* "     @`j < 3 rT1? Rq F>6___PPT9 r* "     @` < 3 r1? q ~v0___PPT106___PPT9 {54 " " #   @`j  < 3 r01?T q F>6___PPT9 r* "     @`  < 3 r1? Tq ~v0___PPT106___PPT9 {24 " " #   @`  < 3 r0#1? q ~v0___PPT106___PPT9 {G4 " " #   @`  < 3 r(1?  P ~v0___PPT106___PPT9 {44 " " #   @`  < 3 r-1?  ~v0___PPT106___PPT9 {04 " " #   @` < 3 r,01?R ~v0___PPT106___PPT9 {34 " " #   @`j < 3 rpB1? R F>6___PPT9 r* "     @`j < 3 rI1?  F>6___PPT9 r* "     @`j < 3 rR1?T  F>6___PPT9 r* "     @`j < 3 r_1? T F>6___PPT9 r* "     @` < 3 rf1?  ~v0___PPT106___PPT9 {F4 " " #   @` < 3 rj1? j P  ~v0___PPT106___PPT9 {34 " " #   @`j < 3 rs1? j  F>6___PPT9 r* "     @` < 3 r$1?Rj  ~v0___PPT106___PPT9 {04 " " #   @` < 3 rԊ1?j R ~v0___PPT106___PPT9 {14 " " #   @` < 3 r$1?j  ~v0___PPT106___PPT9 {44 " " #   @`j < 3 r1?Tj  F>6___PPT9 r* "     @`j < 3 rԞ1?j T F>6___PPT9 r* "     @` < 3 r\1?j  ~v0___PPT106___PPT9 {E4 " " #   @` < 3 r1? P j ~v0___PPT106___PPT9 {24 " " #   @`j < 3 r̹1?  j F>6___PPT9 r* "     @`j < 3 r1?R j F>6___PPT9 r* "     @` < 3 r α1?Rj ~v0___PPT106___PPT9 {04 " " #   @`j  < 3 rxѱ1?j F>6___PPT9 r* "     @`j !< 3 rٱ1?Tj F>6___PPT9 r* "     @`j "< 3 r1?Tj F>6___PPT9 r* "     @` #< 3 r1?j ~v0___PPT106___PPT9 {D4 " " #   @`j $< 3 r81? P F>6___PPT9 r* "     @`j %< 3 r1?  F>6___PPT9 r* "     @`j &< 3 r$1?R F>6___PPT9 r* "     @` '< 3 r41?R~v0___PPT106___PPT9 {24 " " #   @` (< 3 r1?~v0___PPT106___PPT9 {04 " " #   @`j )< 3 r1?TF>6___PPT9 r* "     @`j *< 3 r4%1?TF>6___PPT9 r* "     @` +< 3 r021?~v0___PPT106___PPT9 {C4 " " #   @` ,< 3 r0<1? cP ~v0___PPT106___PPT9 {24 " " #   @` -< 3 rtE1? c ~v0___PPT106___PPT9 {34 " " #   @`j .< 3 rH1?Rc F>6___PPT9 r* "     @`j /< 3 rU1?cRF>6___PPT9 r* "     @` 0< 3 r]1?c~v0___PPT106___PPT9 {24 " " #   @` 1< 3 ra1?Tc~v0___PPT106___PPT9 {04 " " #   @`j 2< 3 rj1?cTF>6___PPT9 r* "     @` 3< 3 rw1?c~v0___PPT106___PPT9 {B4 " " #   @`j 4< 3 r${1? P cF>6___PPT9 r* "     @` 5< 3 r1?  c~v0___PPT106___PPT9 {44 " " #   @`j 6< 3 rx1?R cF>6___PPT9 r* "     @`j 7< 3 rd1?RcF>6___PPT9 r* "     @`j 8< 3 r1?cF>6___PPT9 r* "     @` 9< 3 r1?Tc~v0___PPT106___PPT9 {14 " " #   @` :< 3 r1?Tc~v0___PPT106___PPT9 {04 " " #   @` ;< 3 r1?c~v0___PPT106___PPT9 {A4 " " #   @` << 3 r1? pP ~v0___PPT106___PPT9 {G4 " " #   @` =< 3 rβ1? p ~v0___PPT106___PPT9 {F4 " " #   @` >< 3 rɲ1?Rp ~v0___PPT106___PPT9 {E4 " " #   @` ?< 3 r1?pR~v0___PPT106___PPT9 {D4 " " #   @` @< 3 r(ܲ1?p~v0___PPT106___PPT9 {C4 " " #   @` A< 3 rd1?Tp~v0___PPT106___PPT9 {B4 " " #   @` B< 3 r1?pT~v0___PPT106___PPT9 {A4 " " #   @` C< 3 r1?p P  @`B D<  `1 ?pP pB E<  `1 ?q P q B F<  `1 ?pq B G<  `1 ?P pP q B H<  `1 ?P B I<  `1 ?pq B J<  `1 ?TpTq B K<  `1 ?pq B L<  `1 ?pq B M<  `1 ?RpRq B N<  `1 ? p q B O<  `1 ? p q B P<  `1 ?cP cB Q<  `1 ?P B R<  `1 ?P B S<  `1 ?j P j B T<  `1 ? P  B U<  `1 ? P  V< S ~1?p p  :  b W< C A ?" 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M F>6___PPT9 r* "     @`j  @ 3 rT1?RM F>6___PPT9 r* "     @` !@ 3 r1?MR~v0___PPT106___PPT9 {04 " " #   @`j "@ 3 rh1?MF>6___PPT9 r* "     @`j #@ 3 r1?TMF>6___PPT9 r* "     @`j $@ 3 r!1?MTF>6___PPT9 r* "     @` %@ 3 r.1?M~v0___PPT106___PPT9 {D4 " " #   @`j &@ 3 r21? P MF>6___PPT9 r* "     @`j '@ 3 r8:1?  MF>6___PPT9 r* "     @`j (@ 3 rG1?R MF>6___PPT9 r* "     @` )@ 3 rhO1?RM~v0___PPT106___PPT9 {24 " " #   @` *@ 3 rLY1?M~v0___PPT106___PPT9 {04 " " #   @`j +@ 3 r\1?TMF>6___PPT9 r* "     @`j ,@ 3 rhj1?TMF>6___PPT9 r* "     @` -@ 3 rPr1?M~v0___PPT106___PPT9 {C4 " " #   @` .@ 3 rPv1? P ~v0___PPT106___PPT9 {24 " " #   @` /@ 3 r1?  ~v0___PPT106___PPT9 {34 " " #   @`j 0@ 3 r1?R F>6___PPT9 r* "     @`j 1@ 3 r1?RF>6___PPT9 r* "     @` 2@ 3 rh1?~v0___PPT106___PPT9 {24 " " #   @` 3@ 3 r衵1?T~v0___PPT106___PPT9 {04 " " #   @`j 4@ 3 r1?TF>6___PPT9 r* "     @` 5@ 3 r践1?~v0___PPT106___PPT9 {B4 " " #   @`j 6@ 3 rX1? FP F>6___PPT9 r* "     @` 7@ 3 rȵ1? F ~v0___PPT106___PPT9 {44 " " #   @`j 8@ 3 rD̵1?RF F>6___PPT9 r* "     @`j 9@ 3 r@Ե1?FRF>6___PPT9 r* "     @`j :@ 3 rܵ1?FF>6___PPT9 r* "     @` ;@ 3 rp1?TF~v0___PPT106___PPT9 {14 " " #   @` <@ 3 rp1?FT~v0___PPT106___PPT9 {04 " " #   @` =@ 3 r$1?F~v0___PPT106___PPT9 {A4 " " #   @` >@ 3 r 1? P F~v0___PPT106___PPT9 {G4 " " #   @` ?@ 3 r1? F~v0___PPT106___PPT9 {F4 " " #   @` @@ 3 rd1?R F~v0___PPT106___PPT9 {E4 " " #   @` A@ 3 r#1?RF~v0___PPT106___PPT9 {D4 " " #   @` B@ 3 r-1?F~v0___PPT106___PPT9 {C4 " " #   @` C@ 3 r<61?TF~v0___PPT106___PPT9 {B4 " " #   @` D@ 3 rX?1?TF~v0___PPT106___PPT9  AD "  " #   @` E@ 3 r\C1?F P  @`B F@  `1 ?P B G@  `1 ?P B H@  `1 ?B I@  `1 ?P P B J@  `1 ?FP FB K@  `1 ?B L@  `1 ?TTB M@  `1 ?B N@  `1 ?B O@  `1 ?RRB P@  `1 ? B Q@  `1 ? B R@  `1 ?P B S@  `1 ?P B T@  `1 ?MP MB U@  `1 ?P B V@  `1 ?P B W@  `1 ?TP T5 X@ S ~S1? q % Distances after using A as the pivot2& # $"  & b Y@ C A  ?" s  H @ 0޽h ? 33___PPT10i. ;+D=' *= @B +_u vtntWWDt(  D D S ~]1? :  ;q P  D3 #"2& !  D 3 r,b1? TP ~v0___PPT106___PPT9 {04 " " #   @` D 3 rb1? T ~v0___PPT106___PPT9 {64 " " #   @`j D 3 rHt1?RT F>6___PPT9 r* "     @`j D 3 r|1?TRF>6___PPT9 r* "     @` D 3 r 1?T~v0___PPT106___PPT9 {54 " " #   @`  D 3 r1?TT~v0___PPT106___PPT9 {34 " " #   @`  D 3 r$1?TT~v0___PPT106___PPT9 {24 " " #   @`  D 3 r䚶1?T~v0___PPT106___PPT9 {G4 " " #   @`  D 3 rL1? P T~v0___PPT106___PPT9 {44 " " #   @`  D 3 r1?  T~v0___PPT106___PPT9 {04 " " #   @` D 3 r1?R T~v0___PPT106___PPT9 {34 " " #   @`j D 3 r1?RTF>6___PPT9 r* "     @`j D 3 rD̶1?TF>6___PPT9 r* "     @`j D 3 rԶ1?TTF>6___PPT9 r* "     @`j D 3 r۶1?TTF>6___PPT9 r* "     @` D 3 r1?T~v0___PPT106___PPT9 {F4 " " #   @` D 3 r1? P ~v0___PPT106___PPT9 {34 " " #   @`j D 3 r1?  F>6___PPT9 r* "     @` D 3 r1?R ~v0___PPT106___PPT9 {04 " " #   @` D 3 r1?R~v0___PPT106___PPT9 {14 " " #   @` D 3 r 1?~v0___PPT106___PPT9 {44 " " #   @`j D 3 r1?TF>6___PPT9 r* "     @`j D 3 rx1?TF>6___PPT9 r* "     @` D 3 r+1?~v0___PPT106___PPT9 {E4 " " #   @` D 3 rp/1? MP ~v0___PPT106___PPT9 {24 " " #   @`j D 3 r081? M F>6___PPT9 r* "     @`j D 3 r?1?RM F>6___PPT9 r* "     @` D 3 rL1?MR~v0___PPT106___PPT9 {04 " " #   @`j  D 3 rP1?MF>6___PPT9 r* "     @`j !D 3 rPX1?TMF>6___PPT9 r* "     @`j "D 3 re1?MTF>6___PPT9 r* "     @` #D 3 rm1?M~v0___PPT106___PPT9 {D4 " " #   @`j $D 3 rp1? P MF>6___PPT9 r* "     @`j %D 3 rx1?  MF>6___PPT9 r* "     @`j &D 3 r$1?R MF>6___PPT9 r* "     @` 'D 3 r 1?RM~v0___PPT106___PPT9 {24 " " #   @` (D 3 rX1?M~v0___PPT106___PPT9 {04 " " #   @`j )D 3 rh1?TMF>6___PPT9 r* "     @`j *D 3 r1?TMF>6___PPT9 r* "     @` +D 3 r1?M~v0___PPT106___PPT9 {C4 " " #   @` ,D 3 r1? P ~v0___PPT106___PPT9 {24 " " #   @` -D 3 rH1?  ~v0___PPT106___PPT9 {34 " " #   @`j .D 3 r|Ʒ1?R F>6___PPT9 r* "     @`j /D 3 rͷ1?RF>6___PPT9 r* "     @` 0D 3 r۷1?~v0___PPT106___PPT9 {24 " " #   @` 1D 3 rL߷1?T~v0___PPT106___PPT9 {04 " " #   @`j 2D 3 r\1?TF>6___PPT9 r* "     @` 3D 3 r,1?~v0___PPT106___PPT9 {B4 " " #   @` 4D 3 r1? FP ~v0___PPT106___PPT9 {34 " " #   @` 5D 3 r1? F ~v0___PPT106___PPT9 {44 " " #   @`j 6D 3 r0 1?RF F>6___PPT9 r* "     @`j 7D 3 r1?FRF>6___PPT9 r* "     @` 8D 3 r|!1?F~v0___PPT106___PPT9 {34 " " #   @` 9D 3 r%1?TF~v0___PPT106___PPT9 {14 " " #   @` :D 3 r4/1?FT~v0___PPT106___PPT9 {04 " " #   @` ;D 3 r(81?F~v0___PPT106___PPT9 {A4 " " #   @` D 3 r(P1?R F~v0___PPT106___PPT9 {E4 " " #   @` ?D 3 r\1?RF~v0___PPT106___PPT9 {D4 " " #   @` @D 3 rk1?F~v0___PPT106___PPT9 {C4 " " #   @` AD 3 rt1?TF~v0___PPT106___PPT9 {B4 " " #   @` BD 3 r}1?TF~v0___PPT106___PPT9 {A4 " " #   @` CD 3 rD1?F P  @`B DD  `1 ?P B ED  `1 ?P B FD  `1 ?B GD  `1 ?P P B HD  `1 ?FP FB ID  `1 ?B JD  `1 ?TTB KD  `1 ?B LD  `1 ?B MD  `1 ?RRB ND  `1 ? B OD  `1 ? B PD  `1 ?P B QD  `1 ?P B RD  `1 ?MP MB SD  `1 ?P B TD  `1 ?P B UD  `1 ?TP T1 VD C x1?r x  % Distances after using B as the pivot4& # $"  & b WD C A  ?" s  H D 0޽h ? 33___PPT10i.p<^+D=' *= @B +{ zz WWH'z(  H H S ~1? :  Vw P  H3 #"2&    H 3 r1? TP ~v0___PPT106___PPT9 {04 " " #   @` H 3 rX1? T ~v0___PPT106___PPT9 {64 " " #   @` H 3 r1?RT ~v0___PPT106___PPT9 {94 " " #   @` H 3 r$ĸ1?TR~v0___PPT106___PPT9 {74 " " #   @` H 3 rh1?T~v0___PPT106___PPT9 {54 " " #   @`  H 3 rX͸1?TT~v0___PPT106___PPT9 {34 " " #   @`  H 3 rѸ1?TT~v0___PPT106___PPT9 {24 " " #   @`  H 3 r$ڸ1?T~v0___PPT106___PPT9 {G4 " " #   @`  H 3 r1? P T~v0___PPT106___PPT9 {44 " " #   @`  H 3 r$1?  T~v0___PPT106___PPT9 {04 " " #   @` H 3 r 1?R T~v0___PPT106___PPT9 {34 " " #   @` H 3 r1?RT~v0___PPT106___PPT9 {44 " " #   @` H 3 rx1?T~v0___PPT106___PPT9 {74 " " #   @` H 3 rp!1?TT~v0___PPT106___PPT9 {74 " " #   @` H 3 r1?TT~v0___PPT106___PPT9 {64 " " #   @` H 3 r31?T~v0___PPT106___PPT9 {F4 " " #   @` H 3 r.1? P ~v0___PPT106___PPT9 {34 " " #   @` H 3 rA1?  ~v0___PPT106___PPT9 {94 " " #   @` H 3 r`O1?R ~v0___PPT106___PPT9 {04 " " #   @` H 3 r@J1?R~v0___PPT106___PPT9 {14 " " #   @` H 3 ra1?~v0___PPT106___PPT9 {44 " " #   @` H 3 r\1?T~v0___PPT106___PPT9 {64 " " #   @` H 3 rn1?T~v0___PPT106___PPT9 {54 " " #   @` H 3 rP}1?~v0___PPT106___PPT9 {E4 " " #   @` H 3 r0x1? MP ~v0___PPT106___PPT9 {24 " " #   @` H 3 r1? M ~v0___PPT106___PPT9 {84 " " #   @` H 3 r1?RM ~v0___PPT106___PPT9 |114 " " #   @` H 3 r䜹1?MR~v0___PPT106___PPT9 {04 " " #   @`  H 3 r@1?M~v0___PPT106___PPT9 {74 " " #   @` !H 3 r 1?TM~v0___PPT106___PPT9 {54 " " #   @` "H 3 r1?MT~v0___PPT106___PPT9 {44 " " #   @` #H 3 r1?M~v0___PPT106___PPT9 {D4 " " #   @` $H 3 rʹ1? P M~v0___PPT106___PPT9 {44 " " #   @` %H 3 r0ٹ1?  M~v0___PPT106___PPT9 |104 " " #   @` &H 3 rԹ1?R M~v0___PPT106___PPT9 |134 " " #   @` 'H 3 r|1?RM~v0___PPT106___PPT9 {24 " " #   @` (H 3 r1?M~v0___PPT106___PPT9 {04 " " #   @` )H 3 r1?TM~v0___PPT106___PPT9 {74 " " #   @` *H 3 r`1?TM~v0___PPT106___PPT9 {64 " " #   @` +H 3 r1?M~v0___PPT106___PPT9 {C4 " " #   @` ,H 3 r1? P ~v0___PPT106___PPT9 {24 " " #   @` -H 3 r(1?  ~v0___PPT106___PPT9 {34 " " #   @` .H 3 r-1?R ~v0___PPT106___PPT9 {64 " " #   @` /H 3 r'1?R~v0___PPT106___PPT9 {44 " " #   @` 0H 3 rL:1?~v0___PPT106___PPT9 {24 " " #   @` 1H 3 rPC1?T~v0___PPT106___PPT9 {04 " " #   @` 2H 3 rL1?T~v0___PPT106___PPT9 {44 " " #   @` 3H 3 rU1?~v0___PPT106___PPT9 {B4 " " #   @` 4H 3 r^1? FP ~v0___PPT106___PPT9 {34 " " #   @` 5H 3 rg1? F ~v0___PPT106___PPT9 {44 " " #   @` 6H 3 rp1?RF ~v0___PPT106___PPT9 {74 " " #   @` 7H 3 rz1?FR~v0___PPT106___PPT9 {54 " " #   @` 8H 3 r<1?F~v0___PPT106___PPT9 {34 " " #   @` 9H 3 rT1?TF~v0___PPT106___PPT9 {14 " " #   @` :H 3 rX1?FT~v0___PPT106___PPT9 {04 " " #   @` ;H 3 r1?F~v0___PPT106___PPT9 {A4 " " #   @` H 3 r칻1?R F~v0___PPT106___PPT9 {E4 " " #   @` ?H 3 rû1?RF~v0___PPT106___PPT9 {D4 " " #   @` @H 3 r̻1?F~v0___PPT106___PPT9 {C4 " " #   @` AH 3 rDջ1?TF~v0___PPT106___PPT9 {B4 " " #   @` BH 3 r\޻1?TF~v0___PPT106___PPT9 {A4 " " #   @` CH 3 r1?F P  @`B DH  `1 ?P B EH  `1 ?P B FH  `1 ?B GH  `1 ?P P B HH  `1 ?FP FB IH  `1 ?B JH  `1 ?TTB KH  `1 ?B LH  `1 ?B MH  `1 ?RRB NH  `1 ? B OH  `1 ? B PH  `1 ?P B QH  `1 ?P B RH  `1 ?MP MB SH  `1 ?P B TH  `1 ?P B UH  `1 ?TP T7 VH S ~1?R  % Distances after using G as the pivot4& # $"  & b WH C A  ?" s  H H 0޽h ? 33___PPT10i.𩌌+D=' *= @B +  0L6(  L~ L s *tP   ~ L s *   L0  0e0eA     ? 1 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| &   H L 0޽h ? 33___PPT10i.AH+D=' *= @B +  @P<(  P~ P s *80   ~ P s *   H P 0޽h ? 33___PPT10i. 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Show that a connected, undirected graph has an Euler circuit if and only if each node is of even degree. Let G (V,E) be an undirected graph with m edges in which every node is of even degree. Give an O(V) algorithm to construct an Euler circuit for G. Z(X4  ` 33` Sf3f` 33g` f` www3PP` ZXdbmo` \ғ3y`Ӣ` 3f3ff` 3f3FKf` hk]wwwfܹ` ff>>\`Y{ff` R>&- {p_/̴>?" dd@,|?" dd@   " @ ` n?" dd@   @@``PR    @ ` ` p>> f(    6Q   `}   T Click to edit Master title style! !  0V   `   RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S  0,_  ^ `   >*  0f  ^    @*  0n  ^ `   @*H  0޽h ? 3380___PPT10.,Jm Default Design$  $(  r  S v >   r  S y  `     H  0޽h ? 3380___PPT10.,JmE ld0 (      69 ` 9  j  C 4g @&  "Topics Flow Networks Residual networks Ford-Fulkerson s algorithm Ford-Fulkerson's Algorithm JC0  q]  C  9   x*Further Reading Chapter 25 from Text book2+0 H  0޽h ? ̙33y___PPT10Y+D=' = @B +  @(  ~  s */9       s *49 0 ` vA directed graph can be interpreted as a flow network to analyze material flows through networks. Material courses through a system from a source (where it is produced) to a sink (where it is consumed). Examples : Water through pipelines Newspapers through distribution system Electricity through cables Cars on a production line on roads The source produces the material at a steady rate . The sink consumes the material at a steady rateh0 f0 e0 fd.9  | H  0޽h ? ̙33y___PPT10Y+D=' = @B +  P(  [  00J9  Flow: the rate at which the material moves from one point to another 100 litres of water per hour in a pipe 30 Amperes of electric current in a circuit*0    S hU9 e  k 5 litres/hour 0 4  # ? )TB  C W9 q     S Th9 ; 0M  K30 liters/hour  0   S i9 / j25 litres/hour0 ;  0Xy9   aThe rate at which a material enters a vertex = the rate at which the material leaves the vertex Ba0 0 2aa H  0޽h ? ̙33y___PPT10Y+D=' = @B +     `d (    s *L9 N BThe flow network G =(V,E) is a directed graph in which each edge (u,v) E has a nonnegative capacity c(u,v) 0. If (u,v) E then c(u,v) = 0. A flow network has a source vertex s, and a sink vertex t. For every vertex v V there is a path from s to v and v to t in a connected graph."0 G% *@|B  !       ^2  6jJ 0 X2  0jJP `  X2  0jJ@0X2  0jJ `  X2  0jJ0 p X2  0jJp  jB   BDjJ `0 jB   BDjJ jB   BDjJ  0 jB   BDjJ  jB   BDjJ p  jB  BDjJ `   <$9    Bsource 0   <9  00  @sink 0 jB  BDjJ0   <ܵ9  U:  =s 0   09  @  ?t"0 2H  0޽h ? ̙33y___PPT10Y+D=' = @B +   px(    s *9  xA flow in G is a real-valued function f : V V R that satisfies the following three properties: 1. Capacity constraint : For all u,v V, we require f(u,v) c(u,v). The net flow from one vertex to another must not exceed the given capacity. 2. Skew symmetry : For all u,v V, we require f(u,v) = -f(v,u). The net flow from a vertex u to a vertex v is the negative of the net flow in the reverse direction. The net flow from a vertex to itself is zero for all u V, that is f(u,u) = 0. 3. Flow conservation : For all u V - {s,t}, we require The total net flow out of a vertex other than the source or sink is zero. w0 %6Y  K      k        0  \ f  s *A ?8 X 9  H  0޽h ? ̙33y___PPT10Y+D=' = @B +/  VN (   ~  s *d9  `}  9  L nl Z  # ln Z    B_CDEF1__ @`n    BnC^DE F*1PnOn@^_@`nl     B.CDEF1`..` @`j ZL  lZ  # l Z    B_CDEF1___ @`     BnC^DE F*1nO.^n@` l    B.CDEF1.`..` @` jZL /r m   # r/ m     BOC!DEF1!OO @`L m     BC<DEF1<z @`r    BCDE F*1}3~@`/r f     B~CDEF1~~ @`/ m     BCDEF1 @` L L  rHm   A# U  T    BOC!DEF1OO!O @`{ L m     BC<DEF1<z @` r    BCDE F*1?/03>@` rHf     B~CDEF1~~~ @` Hm     BCDEF1 @` L { H  0޽h ? f̙fy___PPT10Y+D=' = @B +   $t(  $T $ 0t+9  The quantity f(u,v) can be negative or positive, it is called the net flow from vertex u to v. The value of a flow is defined as  0 2.   p f $ s *A ?P  r $ s *> p In the maximum-flow problem, we are given a flow network G with source s and sink t, and we wish to find a flow of maximum value from s to t. There is no net flow between u and v if there is no edge between them. If (u,v) E and (v,u) E, then c(u,v) = c(v,u) = 0. Hence, the capacity constraint, f(u,v) 0 and f(v,u) 0. By skew symmetry, f(u,v) = -f(v,u), therefore, f(u,v) + f(v,u) = 0. Nonzero net flow from vertex u to vertex v implies that (u,v)E or (v,u)E (or both). 0 P E  L        '            A     H $ 0޽h ? ̙33y___PPT10Y+D=' = @B +$  $$66(#(  (R ( s *H8>  Z .Consider the network G=(V,E) shown in the figure below. The network is for a transport system that transports crates of an item from source vertex s to sink vertex t through a number of intermediate points. Each edge (u,v) E in the network is labeled with its capacity c(u,v).D0 6b        ( BDE> R    Qd40  ( BO> ]   Qb40  ( BX>  q Qc40  ( B$b>  Qa40  ( Bdl> h  Q440  ( Bu> ,   R1040   ( Bp> # o  Q940   ( BD> # B  Q740   ( B> /  R1240   ( B> N R2040   ( BԦ> ) w  R1440  ( B>  4  R1340  ( B> D  R1640  ( B>    Q440  ( B> [ 8  Qs40 F2 ( S   F2 ( S 3F2 ( S \x F2 ( S  3G F2 ( S   3 F2 ( S   L o  (# o  ( BCDEFww @`o7Z  ( BCDEFt`tgt @`L  *  (#  *  ( BCDEF @` f  ( BCDEFvgtdv @`)* L |   (#  |  ( BCDEF @`| s   ( BCDEFkvvdk @`.  L 1 K  !(#  1 K  "( BC(DEF( @`1   #( BCDEFCRG @` K L  *  $(#  *  %( BCDEF @` A   &( BCDEFcGrYc @`) * bL ;  '(# ; H (( C S  )( BCDEFVVE @` ;  *( BX>  LH  Qt40 L 1Z  +(# Z1  ,( BC/DEF/ @`1  -( BCDEFARG @` Z bL c  .(#  H /( C d  0( BCDEFTT @`cL c   1(# c   2( B"CDEF" @` F  3( BCDEFRU @`c  L   4(#   5( BCDEF @`  6( BCDEFi`vdi @`%  H ( 0޽h ? ̙33y___PPT10Y+D=' = @B +$  ##66,J#(  , , s *A P0 Let us consider a flow in G, f=19 If f(u,v) >0, edge (u,v) is labeled f(u,v)/c(u,v) If f(u,v) 0, the edge is labeled by its capacity only. 0 0 2@,'              , BPA   Qd40  , BA  (  Qb40  , B(A   Mc00  , B\1A *  Qa40  , B:A  U  S1/440  , BdDA    R1040   , BNA  s o  S4/940   , BWA \ "  S7/740   , BaA [  U12/1240   , BDkA x U15/2040   , BtA  L k  U11/1440  , B~A 4 4@  T8/1340  , BlA  U11/1640  , B$A H x:  S4/440  , BܛA py  Qs40 F2 , S Ԕu^ F2 , S Ԕc F2 , S Ԕc F2 , S ԔX c F2 , S Ԕ@ y F2 , S Ԕ,S L    ,#    , BCDEF @`h  , BCDEFmfmmm @` L   ,#    , BCDEF  @` ` , BCDEF~amj~ @`2L R  j  ,# U  m  , BCDEF  @`R    , BCDEFrnnjr @`4  j L ` " l  !,# % a m  ", BjC+DEF+jj @`` b  #, BCDEFHML @`y " l L  B ^  $,# E a  %, BCDEFv @` ^^  &, BCDEF]Kk_] @`B  bL j U  ',# k X H (, C  '  ), BCDEFQQJ @`i j U  *, BTA   Qt40 L ` }l  +,# a m  ,, BjC2DEF2jj @``   -, BCDEFEMK @`y }l bL  =  .,# 0 3 H /, C 2 J =  0, BCDEFOO @` L N  2  1,# O 5  2, B CDEF  @` w 2  3, BCDEFMO @`N  L J  j  4,# K m  5, BC8DEF&8 @`   6, BCDEFcfojc @`J % j H , 0޽h ? ̙33y___PPT10Y+D=' = @B +  ++0U(  0 0 0A 0 :The positive net flow entering a vertex v is defined by H:0 0 29: f 0 s *A ?   0 0A   XInitially, c(a,b) = 8, and c(b,a) = 3 as shown in Fig. a. f(a,b) = 5 and f(b,a) = 2, the net flow is shown as 3/8 in direction a to b:0 ot         9  0 6tA ) (t  Q340  0 6LA    Q840 :2 0 3 li9   0 6A  c  Qa40 3f3f:2  0 3 lie   0 6A   Qb40 3fPL ) {   0#  ){ <B  0 # lit u   0 BCDEFKK; @`) { PL { \e  0# { \e <B 0 # li e  0 BCDEFII~ @`{ \4  0 6B  w  S2/340  0 6t B    S5/840 :2 0 3 li0 ?  0 6B  ( t Z  Qa40 3f:2 0 3 li[ K  0 6B  2 ~  Ob20 3fPL  K p  0#  K p <B 0 # li   0 BCDEFKK; @` K p PL X p [  0# p X [ <B 0 # li [  0 BCDEFII~ @`X p )  0 6+B ) ;  Q340  0 65B  :  S3/840 :2 0 3 li9 x   0 6?B  ` c  Qa40 3f:2 !0 3 lie  "0 6IB  l  Ob20 3fPL  {  #0#  { <B $0 # li: ;  %0 BCDEFKK; @` { PL  { #e  &0# { #e <B '0 # li e  (0 BCDEFHH~ @` { #4  )0 6UB U3P  SFig.a20  *0 6L_B { 1 SFig.b20  +0 6@iB p 0& SFig.c20 H 0 0޽h ? ̙33y___PPT10Y+D=' = @B +  664o(  4Y 4 0B  ` Af(a,b) = 5 and f(b,a) = 2, the net flow is shown as 5/8 in direction a to b and 2/3 in direction b to a as shown in Fig. b. Then the equivalent flow is 3/8 in the direction a to b as shown in Fig. c. If we increase the flow from b to a from 2 to 6 then the netflow is 1/3 in the direction b to a as shown in Fig. d. >A0 0 2AV     4  4 6$> ^h Q340  4 <LB <6 Q840 :2 4 3 lin; 4 6B % Qa40 3f3f:2 4 3 liG 4 6ԤB : Qb40 3fPL i  4# i<B  4 # li5  4 BCDEFKK; @`iPL    4#  <B  4 # liR+S 4 BCDEFII~ @` i 4 6xB S  S2/340  4 <B <6 S5/840 :2 4 3 lie  4 6B h  Qa40 3f:2 4 3 li  4 64B r  Ob20 3f3fPL   4#  <B 4 # liC D * 4 BCDEFKK; @` PL  ,  4#  , <B 4 # li   4 BCDEFII~ @` , ^ 4 6B ^/ {  Q340  4 <B +z -  S3/840 :2 4 3 lin  4 6B   Qa40 3f:2 4 3 li   4 6xB   Ob20 3f3fPL /   !4# / <B "4 # liz { 5 #4 BCDEFKK; @`/  PL  c  $4#  c <B %4 # li +  &4 BCDEFHH~ @` c i '4 6@ ^ S1/340  (4 < @ <R Q840 :2 )4 3 linR *4 6@ ; Qa40 3f3f:2 +4 3 li^ ,4 6 @ F Ob20 3f3fPL _ -4# _<B .4 # li5 /4 BCDEFKK; @`_PL k 04# k<B 14 # li+ 24 BCDEFII~ @`ki 34 6+@ p SFig.a20  44 65@ 5 f SFig.b20  54 6t?@ < p [ SFig.c20  64 6H@ 8f SFig.d20 H 4 0޽h ? ̙33y___PPT10Y+D=' = @B +]  |8(  8 8 s *hd@  pG (The Ford_Fulkerson method The method is iterative, Starts with f(u,v) for (u,v) V, initial flow of value 0. The method is based on the augmenting path which is defined as a path from s to t along which we can push more flow and then augment flow along this path. Procedure Ford_Fulkerson_method(G,s,t) 1. f 0; 2. while there exists an augmenting path p 3. do augment flow along path p 4. return f j30 a0 0 268|#   .        (    H 8 0޽h ? ̙33y___PPT10Y+D=' = @B +5   \T<(  < < s *]@ z ZResidual Networks Consider a flow network G(V,E) with source s and sink t and let f be a flow in G. Consider a pair of vertices u,v V. Residual capacity between u and v is given by r(u,v) = c(u,v) - f(u,v) the additional net flow we can push from u to v before exceeding the capacity. For example, if c(u,v) = 25 and f(u,v) = 19, then r(u,v) = 6. If f(u,v) < 0 then r(u,v) > c(u,v) Given a flow network G=(V,E) and a flow f, the residual network of G induced by f is Gf=(V,Ef), where Ef ={(u,v) V V : r(u,v) > 0}n0 P n> n0 %0 tMP>z      6      c            X            0HHH < 0޽h ? ̙33y___PPT10Y+D=' = @B +tT  SS||@+S(  @ @ 6@ !v Qd40  @ 6{B { Qb40  @ 6A  Qc40  @ 6ز@ < Qa40  @ 6H@ s S1/440  @ 6@ pj R1040  @ 6@ Y( S4/940   @ 6@   S7/740   @ 6D@ T U12/1240   @ 6@ ^ U15/2040   @ 6@ R U11/1440   @ 6C * T8/1340  @ < C  U11/1640  @ 6C 1@ S4/440  @ 6C `I Qs40 @2 @ C 333li@2 @ C 333li@2 @ C 333li@2 @ C 333li@2 @ C 333lin@2 @ C 333li e L    @#    @ BFCODEFA OF<A @` YS @ BCDEFLRLXL @` L +  @# +  @ zBCDEF  @`\ @ zBCDEFdEMUd @`+ L  @#  @ BMC_DEF D_MQ @`cB @ BCDEF\NNV\ @` L X  @# X !@ zBkC"DEF"kk @`L "@ zBC~DEF~96=~ @`XL  +  #@#  +  $@ zBCqDEFb qb @`  %@ zBCDEFB=LLB @`+ ybL w &@# wH '@ C  (@ BqCDEF9q9; @`w )@ 6@.C |  Qt40 L  *@#  +@ zBkC(DEF(kk @`L ,@ zBC}DEF}86<} @`bL n -@# nH .@ C fn /@ BqCDEFfq77q @`L d 0@# d 1@ zBCDEF @`ed 2@ zBrCDEFr78r @`L  3@#  4@ zBCdDEFV d @`; 5@ zBCDEFFRNUF @`m 6@ 6@ 6(C y' Q440  ?@ 6C  Q540  @@ 6C  & Q340  A@ 6TC % S R1140  B@ 6HC [ R1240  C@ 6ȺC z R R1140  D@ <C Q^  Q540  E@ 6tC K Qt40  F@ 6C p Qd40  G@ 6C  [  Qb40  H@ 6C  ^ Qc40  I@ 6 C Dc  Qa40  J@ 6D   Qs40 @2 K@ C 333u1 @2 L@ C 333u| Q@2 M@ C 333uC$@2 N@ C 333u| @2 O@ C 333uvU@2 P@ C 333u%L  C D Q@# C D R@ zBCDEF  @` ff D S@ zBVCjDEF7jV..7j @`/ C L <: T@# <: U@ zBIC4DEF >4I! @`< V@ zBZCdDEFQZd2/<Q @`M:L    W@#   X@ BCDEFf  @` y  Y@ BVCjDEFfIVj70;I @`@ 9 L  s Z@# s  [@ zBC,DEF, @`  \@ zBRCeDEFeR01e @`sPL l O  ]@# Ol B ^@ 3  O N _@ zBRCcDEF*cR* @`l * L  h `@#  h a@ zBC4DEF4 @`( b@ zBCDEF{UUQ{ @` KhPL  8 v c@# 8 vB d@ 3  w v e@ zBRCdDEFRd(d(CRd @` 8 L   f@#   g@ zB2CzDEF2%i z2 @` ~ h@ zBTCnDEFnTC$< @` 5L  h h i@# h h j@ zBCDEF  @` h H k@ zBWCiDEFiWK(; @` 3 hL  p  l@#  p  m@ zBC!DEF! ! @` p  n@ zBfC}DEFf }%0f @`  L /!f o@# !/f p@ zBCDEF    @`iTf q@ zBCDEF"<BA" @`/!L  1 r@#  1 s@ zBC DEF  @` t@ zBCDEF|QUO| @` K1L  C u@# C  v@ zBC(DEF (  @`  w@ zBCDEF\>IM\ @`C x@ 0D    (Each edge in the residual network can admit positive net flow only. The residual network may include several edges that are not in the original network, (u,v) Ef and (u,v) E is possible (Ef is not a subset of E). However, (u,v) appears in Gf only if (v,u) E and there is a positive flow from v to u. Because the net flow f(u,v) is negative, r(u,v) = c(u,v)-f(u,v) > 0 and (u,v) Ef,0 YB   2 V5@        "      E          ^B y@@ 6Dp^B z@ 6D {@  0e0e    BXCDEF3 Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||aa &0BI#`4p@JX{/&@Q\q Jkil$8Ga7t~~pfWL34@lpsU 4H`sjH=+$& <ZP,c/8K@h@hHtXXCD@                                S" L  |@ # lL+D 1?HAh 74 23H @ 0޽h ? ̙33y___PPT10Y+D=' = @B +  >6D(  D D s *(tD  5 An edge (u,v) can appear in a residual network only if at least one of (u,v) and (v,u) appears in the original network. Ef 2E 0 0 2y|   <    %    D 0D  @m P$Augmenting Paths It is a simple path from s to t in Gf. Each edge (u,v) on an augmenting path admits some additional positive net flow from u to v without violating the capacity constraint on the edge. The residual capacity of a path p is given by, r(p) = min { r(u,v) : (u,v) is in p } r$0 0 2% 5           H D 0޽h ? ̙33y___PPT10Y+D=' = @B +   HH(  H H 0ND  4 0 2f H s *A ?  H 0YD Pp  Let's define a flow function fp,D!0  .    H 0rD   fp is a flow in Gf with value fp= r(p) >0. If we add fp to f, we get another flow in G whose value is closer to the maximum. 0 0 2     G         H H H 0޽h ? ̙33y___PPT10Y+D=' = @B +    0L(  LX L s *أD ` 0Algorithm Procedure Ford-Fulkerson(G,s,t) Input : Flow Network G(V,E) Output : Maximum flow for the given network 1.for each edge (u,v) E 2. do f[u,v] 0; 3. f[v,u] 0; 4.while there exists a path p from s to t in the residual network Gf 5. do r(p) min{r(u,v) : (u,v) is in p}; 6. for each edge (u,v) in p 7. do f[v,u] - f[u,v]; 8. f[u,v] f[u,v] + r(p); 9.return bt0 %0   &H     4  K           G                            0HHH L 0޽h ? ̙33y___PPT10Y+D=' = @B +@  D?(  P P 6KD {( Od20  P 6D ;2 Ob20  P 6D $ Oc20  P 6D V; Oa20  P 6F " O420  P 6 F |) P1020  P 6F ? O920   P 6F s%  O720   P 6t'F rc P1220   P 6,1F N j  P2020   P 6 ;F Z- P1420   P 6DF {  P1320  P 6XNF HL P1620  P 6LXF "" j  O420  P 6aF F~e Os20 @2 P C 333ufQ@2 P C 333u ^G@2 P C 333u @2 P C 333u ^p@2 P C 333u]@2 P C 333u% x GL V; P# ;V P zBiCtDEFe ti^e @`V P zBCDEFU[UbU @``;L 7  P# 7  P zB CDEF   @`  P zBCDEFpLU_p @`z H7 L `I&( P# I`&( P zBqCDEF fqw @``I P zBCDEFfWW_f @`qe&(L \t  P# \t !P BC&DEF& @`\! G "P BCDEF@<C @`tL ;7  #P# ;7  $P BCDEF  @`  %P BCDEFICTTI @`z ;7 PL Ed &P# EdB 'P 3 E0 (P zB}CDEF?}?A @`Cd )P 6rF    Ot20 L \L *P# \L +P BC,DEF, @`\ & ,P BCDEF><C @`LPL 1 -P# 1B .P 3 9K /P zB}CDEF}>>} @`1L q 0P# q 1P zBCUDEFUUU @` 2P zB}CDEF}<>} @`qL K( 3P# K( 4P BCDEF  @` 5P BCDEFN[W^N @`K]( 6P BF < -  Od20  7P BF G   Ob20  8P BpF { Oc20  9P BLF    Oa20  :P B0F  p  P 420  ;P BF   P1020  P B`F 4 G S 4/1220  ?P B@F mI  Q 2020  @P BF    S 4/1420  AP BF  P  Q 1320  BP BlF   R4/1620  CP BG  |+  Q4/420  DP B G k    Os20 F2 EP S |   F2 FP S | F F2 GP S |j F2 HP S | F 2 F2 IP S |   F2 JP S |  L    KP#   LP BC}DEFm }~m @` ?  MP BCDEF]]]d] @`1  L   NP#   OP B>CDEF 2> @`De  PP BCDEFsR]as @`  L   QP#    RP BCDEF  @`  SP BCDEFh__ah @`C ! L D 5  TP#  C 5  UP BC&DEF& @`D 1  VP BCDEFABD @` 5 L    WP#    XP B'CDEF ' @`: B  YP BCDEFOE[WO @`  bL  M  ZP#  M H [P C   D  \P BCDEFEEB @` M  ]P BpG  j=  Ot20 L D i ^P# iC  _P BC-DEF- @`D 1 `P BCDEF?BD @`ibL  m  aP#  m H bP C  k 2  cP BCDEFCC @` m L ,  dP# +  eP BChDEFhhh @`M   fP BCDEFBC @`, L 2   gP# 2   hP BGCDEFG9G @`   iP BCDEFT]^aT @`2  H P 0޽h ? ̙33y___PPT10Y+D=' = @B + k  1j)jPTi(  T)L N Q T#  >1 e T 6,OG * Q540  T 6YG .nQ Q440  T 6bG Ab Q440  T 6lG ~ Q440  T 6vG ;% Qb40  T 6tG 4  Qd40   T 6G & < Qc40   T 6TG Bi Qa40   T 6؜G s  Q440   T 6G i R1040   T 6lG 'W Q440  T 6DG " b W Q740  T 6G \ Q840  T 6G   L R2040  T 6G x= R1040  T 6XG 7O R1340  T 6$G   R1240  T 6G d G  Q440  T 6H  Qs40 H2 T C 333lig5H2 T C 333li[VH2 T C 333li2 H2 T C 333li[$xH2 T C 333li* fH2 T C 333lic + T bD T# dD T zBICSDEFE SI?E @`bD T zBCDEF2d@Z2 @`cAT _X = T#  Xv =  T zBCDEF  @`_X?  !T zBCDEFD-AED @`  =T j 6 "T# u6 #T zBRCeDEF HeRW @`j $T zBCDEF=4AF= @`s 6T ;!e %T# e &T zBrC#DEF#rr @`;/R 'T zBCTDEFT%8(T @`s!eT ul $ (T# ) l{ $ )T zBCyDEFj yj @`l  *T rB}C~DEF;~}h; @`u$XT /$ +T# L$B ,T 3  -T zBKCDEF&K&< @`d/$ .T 6H # N  Qt40 T ;G! /T# G 0T zBrC)DEF)rr @`;e 1T zBCSDEFS"8'S @`sG!XT I 2T# iB 3T 3 , 4T zBJCDEFJ$$J @`IWT  |k 5T# |  6T zBC DEF    @`<S 7T zBKCDEFK#%K @` |k<T LPc 8T#  c 9T zBCqDEFc q @`P  :T zBCDEFV7OHV @`LcT vt ;T# `t T# q ?T zBkC#DEFkk#k @` @T zBCTDEF/Tw, @`3qT -s2# AT# s# BT zBCXDEFJ XJ @`-# CT zBCDEF7q@a7 @`s2T DP! 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