Homework 2 -- due Thursday February 21

Written questions (70 points)

1. [10 points] What are the disadvantages of using a heuristic function for A* which is not consistent? Be precise and provide an example to support the points you are making.
2. [15 points] The functions g and h each play a different role in A*. What are those roles? What happens when you emphasize or de-emphasize one of them by using different weights in f(n)? Consider the case in which f(n) = (1 - w)g(n) + wh(n), with 0 ≤ w ≤ 1. Be specific and analyze what happens for different values of w.
3. [15 points] Discuss the possible advantages of the following state-space search strategy: obtain by some method a path to a goal node and its associated cost f(Goal)=C. This cost is not necessarily minimal but it gives an upper bound on the minimal cost. Now use A* with an admissible $h$ function and discard immediately any OPEN nodes reached whose f values are greater than C.
   1. Explain if the modified A* algorithm with this strategy is guaranteed to find an optimal solution if one exists or not. Be short but precise.
   2. Explain if the fact that the algorithm discards some of the OPEN nodes (i.e. nodes in the frontier) means that fewer nodes are expanded. Be short but precise.
   3. Does this strategy reduce the total storage requirements? Explain your reasoning.
4. [10 points] Answer the following questions on Uniform Cost search briefly but precisely:
   1. Is it possible for Uniform Cost to expand more nodes than Depth-First search? Feel free to use an example to support your answer.
   2. Does Uniform Cost search expand more nodes than A*? Why (or why not)?
5. [20 points] Answer the following questions explaining your reasoning briefly but precisely.
   1. Why any node in OPEN with f(n) < f^*(n) (the cost of the optimal solution path) will eventually be selected for expansion by A*?
   2. Is it true that all admissible heuristics are equal in the sense that A* will search the states in the same order no matter what the heuristic is?
   3. Is Breadth-First search complete if the state space has infinite depth but a finite branching factor?
   4. Does the fact that A* is ``optimally efficient'' mean that A* will never expand more nodes than any other algorithm?

Programming questions (30 points)

1. [5 points] Compare the performance of the following uninformed search algorithms:
   1. breadth_first_tree_search,
   2. breadth_first_graph_search,
   3. depth_first_graph_search,
   4. iterative_deepening_search,
   5. depth_limited_search
   on the Romanian map to solve these problems:
   1. find a path from Arad to Bucharest
   2. find a path from Oradea to Neamt
   3. find a path from Eforie to Timisoara
   To compare the performance, you can use the Lisp function compare-search-algorithms or the python function compare_graph_searchers().
2. [10 points] Add or modify the code in the search functions to compute the cost of the solution found by each algorithm and print the cost for each algorithm when you compare the algorithms.
3. [5 points] Print the solution path found by the search functions.
4. [10 points] Compute the average cost and standard deviation of the cost of the paths found by the different algorithms you used in the first programming question when you use them to solve the same problem. Write the code so that it will work for any number of algorithms and will print the average and standard deviation at the bottom of the table printed by the comparison function.