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CS 440: INTRODUCTION TO ARTIFICIAL INTELLIGENCE FALL 2021

Assignment 2

Search Problems in AI

Deadline: November 12th, 11:55pm.

Perfect score: 70.

Assignment Instructions:

Teams: Assignments should be completed by teams of up to three students. You can work on this assignment individually

if you prefer. No additional credit will be given for students that complete an assignment individually. Make sure to write

the name and RUID of every member of your group on your submitted report.

Submission Rules: Submit your reports electronically as a PDF document through Canvas (canvas.rutgers.edu).

Each team of students should submit only a single copy of their solutions and indicate all team members on their

submission. Failure to follow these rules will result in lower grade for this assignment.

Late Submissions: No late submission is allowed. 0 points for late assignments.

Extra Credit for LATEX: You will receive 10% extra credit points if you submit your answers as a typeset PDF (using

LATEX, in which case you should also submit electronically your source code). There will be a 5% bonus for electronically

prepared answers (e.g., on MS Word, etc.) that are not typeset. If you want to submit a handwritten report, scan it and

submit a PDF via Canvas. We will not accept hardcopies. If you choose to submit handwritten answers and we are not able

to read them, you will not be awarded any points for the part of the solution that is unreadable.

Precision: Try to be precise. Have in mind that you are trying to convince a very skeptical reader (and computer scientists

are the worst kind...) that your answers are correct.

Collusion, Plagiarism, etc.: Each team must prepare its solutions independently from other teams, i.e., without using

common notes, code or worksheets with other students or trying to solve problems in collaboration with other teams. You

must indicate any external sources you have used in the preparation of your solution. Do not plagiarize online sources and

in general make sure you do not violate any of the academic standards of the department or the university. Failure to follow

these rules may result in failure in the course.

Problem 1 (10 points): Trace the operation of A? search (the tree version) applied to the problem of getting to Bucharest

from Lugoj using the straight-line distance heuristic. That is, show the sequence of nodes that the algorithm will con-

sider and the f , g, and h score for each node. You don’t need to draw the graph, just right down a sequence of

Figure 1: A simplified road map of part of Romania indicating distances between different cities.

Section 3.5. Informed (Heuristic) Search Strategies 93

U

Figure 3.22 Values of hSLD—straight-line distances to Bucharest.

expanding a node that is not on the solution path; hence, its search cost is minimal. It is

not optimal, however: the path via Sibiu and Fagaras to Bucharest is 32 kilometers longer

than the path through Rimnicu Vilcea and Pitesti. This shows why the algorithm is called

“greedy”—at each step it tries to get as close to the goal as it can.

Greedy best-first tree search is also incomplete even in a finite state space, much like

depth-first search. Consider the problem of getting from Iasi to Fagaras. The heuristic sug-

gests that Neamt be expanded first because it is closest to Fagaras, but it is a dead end. The

solution is to go first to Vaslui—a step that is actually farther from the goal according to

the heuristic—and then to continue to Urziceni, Bucharest, and Fagaras. The algorithm will

never find this solution, however, because expanding Neamt puts Iasi back into the frontier,

Iasi is closer to Fagaras than Vaslui is, and so Iasi will be expanded again, leading to an infi-

nite loop. (The graph search version is complete in finite spaces, but not in infinite ones.) The

worst-case time and space complexity for the tree version is O(bm), where m is the maximum

depth of the search space. With a good heuristic function, however, the complexity can be

reduced substantially. The amount of the reduction depends on the particular problem and on

the quality of the heuristic.

3.5.2 A* search: Minimizing the total estimated solution cost

The most widely known form of best-first search is called A? search (pronounced “A-starA? SEARCH

search”). It evaluates nodes by combining g(n), the cost to reach the node, and h(n), the cost

to get from the node to the goal:

f(n) = g(n) + h(n) .

Since g(n) gives the path cost from the start node to node n, and h(n) is the estimated cost

of the cheapest path from n to the goal, we have

f(n) = estimated cost of the cheapest solution through n .

Thus, if we are trying to find the cheapest solution, a reasonable thing to try first is the

node with the lowest value of g(n) + h(n). It turns out that this strategy is more than just

reasonable: provided that the heuristic function h(n) satisfies certain conditions, A? search is

both complete and optimal. The algorithm is identical to UNIFORM-COST-SEARCH except

that A? uses g + h instead of g.

Figure 2: Straight-line distances to Bucharest

Problem 2 (10 points): Consider a state space where the start state is number 1 and each state k has two successors:

numbers 2k and 2k + 1.

(a) Suppose the goal state is 11. List the order in which states will be visited for breadthfirst search, depth-limited search

with limit 3, and iterative deepening search.

(b) How well would bidirectional search work on this problem? List the order in which states will be visited. What is the

branching factor in each direction of the bidirectional search?

Problem 3 (5 points): Which of the following statements are correct and which ones are wrong?

(a) Breadth-first search is a special case of uniform-cost search.

(b) Depth-first search is a special case of best-first tree search.

(c) Uniform-cost search is a special case of A? search.

(d) Depth-first graph search is guaranteed to return an optimal solution.

(e) Breadth-first graph search is guaranteed to return an optimal solution.

(f) Uniform-cost graph search is guaran e d to return an optimal solution.

(g) A? graph search is guaranteed to return an optimal solution if the heuristic is consistent.

(h) A? graph search is guaranteed to expand no more nodes than depth-first graph search if the heuristic is consistent.

(i) A? graph search is guaranteed to expand no more nodes than uniform-cost graph search if the heuristic is consistent.

Problem 4 (2 points): Iterative deepening is sometimes used as an alternative to breadth first search. Give one advantage

of iterative deepening over BFS, and give one disadvantage of iterative deepening as compared with BFS. Be concise and

specific.

Problem 5 (10 points): Prove that if a heuristic is consistent, it must be admissible. Construct an example of an admissible

heuristic that is not consistent. (Hint: you can draw a small graph of 3 nodes and write arbitrary cost and heuristic values

so that the heuristic is admissible but not consistent).

Problem 6 (3 points): In a Constraint Satisfaction Problem (CSP) search, explain why it is a good heuristic to choose the

variable that is most constrained but the value that is least constraining.

Problem 7 (10 points): Consider the following game tree, where the first move is made by the MAX player and the second

move is made by the MIN player.

(a) What is the best move for the MAX player using the minimax procedure?

(b) Perform a left-to-right (left branch first, then right branch) alpha-beta pruning on the tree. That is, draw only the parts

of the tree that are visited and don’t draw branches that are cut off (no need to show the alpha or beta values).

(c) Do the same thing as in the previous question, but with a right-to-left ordering of the actions. Discuss why different

pruning occurs.

Alpha-Beta Pruning

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