Consider the navigation problem shown in Figure 1. The number next to
each
edge is the cost of the performing the action corresponding to that
edge. You start from A and your goal is to get to F. List the
order in which nodes are expanded,
which nodes are added to the fringe
and which states are added to the
closed set when performing Graph
Search using:
breadth-first search.
depth-first search.
iterative deepening search.
uniform cost search.
greedy search
A* search.
Figure 1: Search Task for Task 1
Note for greedy and A* search use the following heuristic:
h(A) = 12
h(B) = 10
h(C) = 10
h(D) = 3
h(E) = 3
h(F) = 0
Task 2
A social network graph (SNG) is
a graph where each vertex is a
person and each edge represents an acquaintance. In other words, an SNG
is a graph showing who knows who. For example, in the graph shown on
Figure 2, George knows Mary and John, Mary knows Christine, Peter and
George, John knows Christine, Helen and George, Christine knows Mary
and John, Helen knows John, Peter knows Mary.
The degrees of separation measure how closely connected two people are
in the graph. For example, John has 0 degrees of separation from
himself, 1 degree of separation from Christine, 2 degrees of separation
from Mary, and 3 degrees of separation from Peter.
From among general tree search
using breadth-first search,
depth-first search,
iterative deepening search, and uniform cost search, which one(s)
guarantee finding the correct number of degrees of separation between
any two people in the graph?
For the SNG shown in Figure 2, draw the
first three
levels of the search tree, with John as the starting point (the first
level of the tree is the root).
Is there a one-to-one correspondence
between nodes in the search tree and vertices in the SNG (i.e. does
every node in the search tree correspond to a vertex in the SNG)? Why,
or why
not? In your answer here, you should assume that the search algorithm
does not try to avoid revisiting the same state.
Draw an SNG containing exactly 5 people,
where at least
two people have 4 degrees of separation between them.
Draw an SNG containing exactly 5 people,
where all
people have 1 degree of separation between them.
In an implementation of
breadth-first tree
search for
finding degrees of separation, suppose that every node in the search
tree takes 1KB of memory. Suppose that the SNG contains one million
people. Outline (briefly but precisely) how to make sure that the
memory required to store search tree nodes will not exceed 1GB (the
correct answer can be described in one-two lines of text). In your
answer here you are free to enhance/modify the breadth-first search
implementation as you wish, as long as it remains breadth-first (a
modification that, for example, converts breadth-first search into
depth-first search or iterative deepening search is not
allowed).
Figure 2: A Social Network Graph
Task 3
Figure
3: A search problem showing states and costs of moving from one state
to
another. Costs are undirected.
Consider
the search space shown in Figure 3. D is the only goal state. Costs are
undirected. For each of the following heuristics, determine if it is
admissible or not. For non-admissible heuristics, modify their values
as needed to make them admissible.
Consider a search space, where each state can be red, green,
blue, yellow, or black. Multiple states may have the same color. The
goal is to reach any black state. Here are some rules on the successors
of different states, based on their color (these successor functions
are unidirectional):
Red states can only have green or blue children.
Blue states can only have red or black children.
Green states can only have blue or yellow children.
Yellow states can only have yellow or red children.
Black states can only have green or black children.
Define a maximally admissible heuristic that assigns a value to each
state based only on the color of that state
Assume that any move from one state to another has cost
1.
Task 5
Figures 4 and 5 show maps where
all the towns are on a
grid. Each town T has coordinates (Ti, Tj), where Ti Tj are
non-negative integers. We use the term Euclidean distance for the
straight-line distance between two towns, and the term driving distance
for the length of the shortest driving route connecting two towns. The
only roads that exist connect towns that have Euclidean (straight-line)
distance 1 from each other (however, there may be towns with Euclidean
distance 1 from each other that are NOT directly connected by a road,
for example in Figure 5).
Consider greedy search, where the node to be expanded is always the one
with the shortest Euclidean distance to the destination. Also consider
A* search, where h(n) is the Euclidean distance from n to the
destination (remember that the next node is picked not based on h(n)
but based on f(n) = g(n) + h(n)). For each of the maps showing on
Figures 4 and 5, which of the following statements is true?
Greedy search always performs better than or the same as A*.
Greedy search always performs worse than or the same as A*.
Greedy search performs sometimes better, sometimes worse,
and sometimes
the same as A*, depending on the start and end states.
Justify your answer. For the purposes of this question, the performance
of a search algorithm is simply measured by the number of nodes visited
by that algorithm. Note that you have
to provide separate answers for
Figure 4 and for Figure 5.
Figure
4: A map of cities on a fully connected grid. Every city is
simply named by its coordinates.
Figure 5: A map of cities on a partially connected grid. Every city is
simply named by its coordinates.
