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

Info iconThis preview shows pages 1–11. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 2
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
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 4
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
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 6
Example Example strategy for Min: strategy for Max: Max strategy Min strategy mixed mixed
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 8
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
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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.
Background image of page 10
Image of page 11
This is the end of the preview. Sign up to access the rest of the document.

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.

Page1 / 81

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

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online