cs188_sp09_mt1_sol 6

# cs188_sp09_mt1_sol 6 - 6 4(10 points Multi-agent Search...

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Unformatted text preview: 6 4. (10 points) Multi-agent Search: Connect-3 In the game of Connect-3, players X and O alternate moves, dropping their symbols into columns 1, 2, 3, or 4. Three-in-a-row wins the game, horizontally, vertically or diagonally. X plays first. X O X 1 O X O X O X O X X O 2 1 3 X O X O O X 3 2 4 X O O X 4 3 O X O O X O 4 1 X O X X 2 O 3 O X 4 X 1 2 1 2 4 (a) (1 pt) What is the maximum branching factor of3minimax search for Connect-3? 4. (b) (1 pt) What is the maximum tree depth in plies? 6. According to our terminology, a search ply in a multi-agent search problem includes one move for every agent. Because the textbook differs in its definition, we also accepted 12. (c) (1 pt) Give a reasonably tight upper bound on the number of terminal states. Several answers were accepted: 412 is a reasonable bound on the number of leaves of the search tree. 312 is a reasonable bound on the number of total states in the game, because each square can be empty, contain an X or contain an O. A tighter bound is 154 , which follows from the observation that there are only 15 ways to legally fill a column from the bottom up with X's and O's. Bounds based on the assumption that the entire board would be filled with X's and O's were not accepted, even if they in fact overestimated the true number of terminal states. (d) (2 pt) Draw the game tree starting from the board shown above right (with O to play next). You may abbreviate states by drawing only the upper-right region circled. The root node is drawn for you. X O X X O X 1 X O X 1 X O X 1 O X X 2 O X X 2 O X X O O X X 2 !"# O 3 O X 4 X O X 1 X O O X X O 1 3 X 2 !"# 1 O O X X O \$# O O O 3 X X 4 O X X X O 2 X O 1 X O X X 2 4 O 3 O X 4 O O 3 XO X XX O 24 X O X 1 X X O O X X O X 4 O X O O X X 1 4 O X 3 O O X X O 2 X O 4 X O O X X 2 4 X O 3 O X 4 O X 3 1 X X O O X 3 X X 2 O XO O 13 X X 4 O X 1 !"# 4 2 O O X O X O X O 1 3 O 3 O X O 4 3 O X X O \$# !%# O X 3 1 X O X 1 X O X O X X 3 2 1 X O X X O X 3 X O 2 O X O X 3 2 X !"# X O X X 2 1 4 2 X O 2 X O O X 2 4 X 4 1 O X O 4 X !%# O O 3 X X X O 4 1 3 O 4 (e) (2 pt) X is the maximizer, while O is the minimizer. X's utility for terminal states is k when X wins and 1 2 3 4 1 2 3 4 -k when O wins, where k is 1 for a horizontal 3-in-a-row win (as in above left), 2 for a vertical win, and 3 for a diagonal win. A tie has value 0. Label each node of your tree with its minimax value. 4 3 2 X 2 1 X (f ) (3 pt) Circle all nodes of your tree that will not be explored when using alpha-beta pruning and a move ordering that maximizes the number of nodes pruned. ...
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