Homework Assignment 2 Solution

Problem 1: Minimax and Alpha-Beta [15 points]




(a) [4] Use the Minimax algorithm to compute the minimax value at each node for the game tree below.



























































































[9] Use the Alpha-Beta pruning algorithm to prune the game tree above assuming child nodes are visited from left to right. Show all final alpha and beta values computed at the root, each internal node visited, and at the top of pruned branches. Note: Follow the algorithm in Figure 5.7 in the textbook. Also show the pruned branches.



[2] For a general game tree (i.e., not limited to the above tree), are there any cases that the Alpha-Beta algorithm gives a different value at the root node than the Minimax algorithm? If yes, show an example; if no, just say no.






Problem 2: Hill-Climbing [20 points]




Given a set of locations and distances between them, the goal of the Traveling Salesperson Problem (TSP) is to find the shortest tour that visits each location exactly once. Assume that you do not return to the start location after visiting the last location in a tour. We would like to solve the TSP problem using a greedy hill-climbing algorithm. Each state corresponds to a permutation of all the locations (called a tour). The operator neighbors(s ) generates all neighboring states of state s by swapping two locations. For example, if s = <A-B-C is a tour, then <B-A-C, <C-B-A and <A-C-B are the three neighbors generated by neighbors(s). Use as the evaluation function for a state the length of the tour, where each pairwise distance is given in a distance matrix. For example, if s = <A-B-C is a tour, then its total length is d(A, B)




d(B, C) where d(A, B) is the distance between location A and B.



[3] If there are n locations, how many neighboring states does the neighbors(s) function produce?



[3] What is the total size of the entire search space, i.e., the total number of states, when there are n locations?



[14] Imagine that a student wants to hand out fliers about an upcoming programming contest. The student wants to visit the Memorial Union (M), Wisconsin Institute of Discovery (W), Computer Sciences Building (S), and Engineering Hall (E) to deliver the fliers. The goal is to find a tour as short as possible. The distance matrix between these locations is given as follows:








M
W
E
S










M
0
1.1
1.4
0.9










W
1.1
0
0.6
0.7










E
1.4
0.6
0
0.5










S
0.9
0.7
0.5
0













The student starts applying the hill-climbing algorithm from the initial state: <W-M-E-S




[2] What is the length of the tour associated with the initial state?



[4] What are the possible neighboring states of <W-M-E-S and what are each of their tour lengths?



[2] What is the next state reached by hill-climbing? Or explain why there is no next state, if there isn’t one.



[6] If (iii) had a next state, continue the algorithm until it terminates, and list the sequence of states found, from initial state to final state. What is the tour length associated with the final state?












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CS 540 Fall 2019







Problem 3: Take-Stones Game Playing [65 points]




Implement the Alpha-Beta pruning algorithm to play a two-player game called Take-Stones. Follow the algorithm pseudocode described in Figure 5.7 of the textbook. Different versions of the Alpha-Beta algorithm may lead to different values of alpha and beta.




Game Rules

The game starts with n stones numbered 1, 2, 3, ..., n. Players take turns removing one of the remaining numbered stones. At a given turn there are some restrictions on which numbers (i.e., stones) are legal candidates to be taken. The restrictions are:




At the first move, the first player must choose an odd-numbered stone that is strictly less than n/2. For example, if n = 7 (n/2 = 3.5), the legal numbers for the first move are 1 and 3. If n = 6 (n/2 = 3), the only legal number for the first move is 1.



At subsequent moves, players alternate turns. The stone number that a player can take must be a multiple or factor of the last move (note: 1 is a factor of all other numbers). Also, this number may not be one of those that has already been taken. After a stone is taken, the number is saved as the new last move. If a player cannot take a stone, he/she loses the game.



An example game is given below for n = 7:




Player 1: 3




Player 2: 6




Player 1: 2




Player 2: 4




Player 1: 1




Player 2: 7




Winner: Player 2







Program Specifications




There are 2 players: player 1 (called Max) and player 2 (called Min) . For a new game (i.e., no stones have been taken yet), the Max player always plays first. Given a specific game board state, your program is to compute the best move for the current player using the given heuristic static board evaluation function. That is, only a single move is computed.




Input




A sequence of positive integers given as command line arguments separated by spaces:




java Player <#stones <#taken_stones <list_of_taken_stones <depth




#stones: the total number of stones in the game



#taken_stones: the number of stones that have already been taken in previous moves. If



this number is 0, this is the first move in a game, which will be played by Max. (Note: If this number is even, then the current move is Max’s; if odd, the current move is Min’s)




List_of_taken_stones: a sequence of integers indicating the indexes of the already-taken stones, ordered from first to last stone taken. Hence, the last stone in the list was the stone taken






in the last move. If #taken_stones is 0, this list will be empty. You can safely assume that these stones were taken according to the game rules.




depth: the search depth. If depth is 0, search to end game states (i.e., states where a winner is determined).



For example, with input: java Player 7 2 3 6 0, you have 7 stones while 2 stones, numbered 3 and 6, have already been taken, so it is the Max player’s turn (because 2 stones have been taken), and you should search to end game states.




Output




You are required to print out the following information to the console, after your alpha-beta search has completed:




The best move (i.e., the stone number that is to be taken) for the current player (as computed by your alpha-beta algorithm)



The value associated with the move (as computed by your alpha-beta algorithm)



The total number of nodes visited



The number of nodes evaluated (either an end game state or the specified depth is reached)



The maximum depth reached (the root is at depth 0)



The average effective branching factor (i.e., the average number of successors that are not pruned)



For example, here is sample input and output when it is Min’s turn to move (because 3 stones have previously been taken), there are 4 stones remaining (3 5 6 7), and the Alpha-Beta algorithm should generate a search tree to maximum depth 3. Since the last move was 2 before starting the search for the best move here, only one child is generated corresponding to removing stone 6 (since it is the only multiplier of 2). That child node will itself have only one child corresponding to removing stone 3 (since it is the only factor of 6 among the remaining stones) . So, the search tree generated will have the root node followed by taking 6, followed by taking 3, which leads to a terminal state (Max wins). So, returning from these nodes, from leaf to root, we get the output below.




Input:

$java TakeStones 7 3 1 4 2 3




Output:

Move: 6

Value: 1.0

Number of Nodes Visited: 3

Number of Nodes Evaluated: 1

Max Depth Reached: 2

Avg Effective Branching Factor: 1.0




All doubles/floats should be printed with 1 decimal place.




Static Board Evaluation




The static board evaluation function should return values as follows:




At an end game state where Player 1 (MAX) wins: 1.0



At an end game state where Player 2 (MIN) wins: -1.0






Otherwise,



if it is Player 1 (MAX)’s turn:

If stone 1 is not taken yet, return a value of 0 (because the current state is a relatively neutral one for both players)



If the last move was 1, count the number of the possible successors (i.e., legal moves). If the count is odd, return 0.5; otherwise, return -0.5.



If last move is a prime, count the multiples of that prime in all possible successors. If the count is odd, return 0.7; otherwise, return -0.7.



If the last move is a composite number (i.e., not prime), find the largest prime that can divide last move, count the multiples of that prime, including the prime number itself if it hasn’t already been taken, in all the possible



successors. If the count is odd, return 0.6; otherwise, return -0.6.




If it is Player 2 (MIN)’s turn, perform the same checks as above, but return the opposite (negation) of the values specified above.



Other Important Information

Set the initial value of alpha to be Double.NEGATIVE_INFINITY and beta to be
Double.POSITIVE_INFINITY




To break ties (if any) between child nodes that have equal values, pick the smaller numbered stone. For example, if stones 3 and 7 have the same value, pick stone 3.



Your program should run within 10 seconds for each test case.



The Skeleton Code

You can use the provided skeleton code or feel free to implement your program entirely on your own.




However, you should ensure that your program:




Has a main class named TakeStones (we will call TakeStones to test)



Receives input from the command line as specified



Outputs exactly match our given format



If you use the skeleton, you should implement the two methods in class GameState, two methods in class Helper, and three methods in class AlphaBetaPruning. You may also need to add some additional class fields or helper methods.




Class GameState: defines the state of a game




size: the number of stones



stones: a Boolean array of stones, a false value indicates that that stone has been taken. Notice that 0 is not a stone, so stones[0] = false, and stones has a size of size + 1



lastMove: index of the last taken stone
Class TakeStones: the main function that parses the command line inputs and runs the search algorithm




Class AlphaBetaPurning: where the Alpha-Beta pruning algorithm should be implemented




Class Helpers: helper functions




More details can be found in the comments in the supplied code.










There are three more test cases in the testcases folder to help verify your implementation. The testcase.txt file contains the command line arguments for the three cases, and the correct outputs are in files output[1-3].txt




Deliverables

Put all .java files needed to run your program, including ones you wrote, modified or were given and are unchanged, into a folder called <wiscNetID-HW2-P3 Compress this folder to create <wiscNetID-HW2-P3.zip and upload it to Canvas. For example, for someone with UW NetID crdyer@wisc.edu the file name must be: crdyer-HW2-P3.zip You should not include any package path or external libraries in your program.


























































































































































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