Assignment 2

Written Assignment - Uninformed & Informed Search

Max points:
The assignment should be submitted via Blackboard.

Instructions

Question 1

Max: [4308: 20 Points, 5360: 16 Points]

Consider the search tree 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 the node A. The goal is the reach node G. List the order in which nodes will be visited using:
Figure 1: Search Tree
Figure 1: Search Tree for Problem 1

Question 2

Max: [4308: 20 Points (+5 points EC), 5360: 25 Points]

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.
  1. 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 (assume we are using the stategies without any modifications)?

  2. 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.

  3. Draw an SNG containing exactly 5 people, where at least two people have 4 degrees of separation between them.

  4. Draw an SNG containing exactly 5 people, where all people have 1 degree of separation between them.

  5. CSE 5360 Only (5 point EC for CSE 4308): 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
Figure 2: A Social Network Graph

Question 3

Max: [4308: 15 Points, 5360: 9 Points]

Figure 5. A search graph showing states and costs of moving from one state to another. Costs are undirected.
Figure 3. A search graph 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.

Heuristic 1:
      h(A) = 50
      h(B) = 35
      h(C) = 5
      h(D) = 0
      h(E) = 45
      h(F) = 10

Heuristic 2:
      h(A) = 70
      h(B) = 70
      h(C) = 70
      h(D) = 70
      h(E) = 70
      h(F) = 70

Heuristic 3:
      h(A) = 40
      h(B) = 20
      h(C) = 5
      h(D) = 0
      h(E) = 5
      h(F) = 20

Question 4

Max: [4308: 25 Points, 5360: 15 Points]

Consider a search space, where each state can be a city, suburb, farmland, or mountain. The goal is to reach any state that is a mountain. Here are some rules on the successors of different states:
Define the best admissible heuristic h you can define using only the above information (you should not assume knowledge of any additional information about the state space). By "best admissible" we mean that h(n) is always the highest possible value we can give, while ensuring that heuristic h is still admissible.

You should assume that every move from one state to another has cost 1.

Question 5

Max: [4308: 20 Points, 5360: 15 Points]

Figure 6. An example of a start state (left) and the goal state (right) for the 24-puzzle.
Figure 4. An example of a start state (left) and the goal state (right) for the 24-puzzle.

The 24-puzzle is an extension of the 8-puzzle, where there are 24 pieces, labeled with the numbers from 1 to 24, placed on a 5x5 grid. At each move, a tile can move up, down, left, or right, but only if the destination location is currently empty. For example, in the start state shown above, there are three legal moves: the 12 can move down, the 22 can move left, or the 19 can move left. The goal is to achieve the goal state shown above. The cost of a solution is the number of moves it takes to achieve that solution.

For some initial states, the shortest solution is longer than 100 moves. 
For all initial states, the shortest solution is at most 208 moves.

An additional constraint is that, in any implementation, storing a search node takes 1000 bytes, i.e., 1KB of memory.

Consider general tree search using the stategies of breadth-first search, depth-first search, iterative deepening search and uniform cost search.

(a): Which (if any), among those methods, can guarantee that you will never need more than 50KB of memory to store search nodes? Briefly justify your answer.

(b): Which (if any), among those methods, can guarantee that you will never need more than 1200KB of memory to store search nodes? Briefly justify your answer.

Question 6 (Extra Credit for 4308, Required for 5360)

Max: [4308: 20 Points EC, 5360: 20 Points]

Figures 5 and 6 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 4).

Consider greedy search, where the node to be expanded is always the one with the shortest Euclidean distance to the destination [h(n) where h(n) is the Euclidean distance from n to the destination]. Also consider A* search where the next node to expand is picked f(n) = g(n) + h(n) [when g(n) is cumulative actual road distance between start and n]. For each of the maps showing on Figures 3 and 4, which of the following statements is true?
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 5 and for Figure 6.

Figure 3: Fully connected city grid
Figure 5. A map of cities on a fully connected grid. Every city is simply named by its coordinates.

Figure 4: Partialy connected city grid
Figure 6. A map of cities on a partially connected grid. Every city is simply named by its coordinates.