# new8-11 - Performance analysis of Alpha-Beta Pruning...

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Performance analysis of Performance analysis of Alpha-Beta Pruning Alpha-Beta Pruning Since alpha-beta pruning performs a minimax search while pruning much of the tree, its effect is to allow a deeper search with the same amount of computation. The question: how much does alpha-beta improve performance? The best way to characterize is asymptotic effective asymptotic effective branching factor. branching factor. The dth root of the number of nodes (in a search to depth d, in the limit of large d) number of nodes generated at depth d / number of nodes generated at depth d-1.
The efficiency of alpha-beta pruning depends upon the order in which nodes are encountered at the search frontier. Thus, we consider 3 different cases: worst case - the algorithm doesn’t perform any cutoffs at all best case average case Performance analysis of Performance analysis of Alpha-Beta Pruning Alpha-Beta Pruning

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Example of alpha-beta worst case Example of alpha-beta worst case Evaluation from left to right 4 4 14 14 13 12 11 2 10 1 9 8 7 6 2 3 5 14 12 2 1 8 6 2 4 14 2 8 4 2 4 MAX MIN
Lower Bound for Minimax Lower Bound for Minimax Algorithms Algorithms We consider a lower bound on the number of leaf nodes that must be examined by any minimax algorithm. In minimax algorithm, it’s a guaranty to return the minimax value v of the root node of a game tree. verifying maximum value = v verifying value v && value v . Any correct minimax algorithm must explore: a strategy for Max a strategy for Min

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Strategies for Min and Max Strategies for Min and Max value value v: v: doesn’t matter what min does Strategy for max Strategy for max : subtree containing: one child of each Max node all b children of each min node value value v: v: doesn’t matter what Max does Strategy for min Strategy for min : subtree containing: one child of each Min node all b children of each Max node
Example Example strategy for Min: strategy for Max: Max strategy Min strategy mixed mixed

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Strategy for Max Strategy for Max d is even leaf nodes d is odd leaf nodes Strategy for Min Strategy for Min d is even leaf nodes d is odd leaf nodes b d 2 b d 2 b d 2 b d 2 Assume : uniform branching factor of b uniform depth of d levels Max move is at the root. Lower Bound for Minimax Lower Bound for Minimax Algorithms - Analysis Algorithms - Analysis
Total number of distinct leaf nodes Total number of distinct leaf nodes : d is odd : d is even : note: note: there is a single leaf node in common of both strategies. Lower Bound for Minimax Lower Bound for Minimax Algorithms - Analysis Algorithms - Analysis     b + b b + b d/2 d/2 d/2 d/2 = b + b d/2 d/2

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d/2 + b d/2 -1 = O(b d/2 ) b d/2 + b d/2 -1 = O(b d/2 ) This is the number of leaf nodes that must be examined by any minimax algorithm. This is the lower bound of the time complexity.
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## This note was uploaded on 10/23/2011 for the course ENCS ENCS5 taught by Professor Abdelsalam during the Spring '10 term at Birzeit University.

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new8-11 - Performance analysis of Alpha-Beta Pruning...

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