# The solution for the branching factor of the alpha-beta pruning algorithm

Session 16: S. Even, Chairman

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## Abstract

This paper analyzes N_{n,d}, the average number of terminal nodes examined by the α-β pruning algorithm in a uniform game-tree of degree n and depth d for which the terminal values are drawn at random from a continuous distribution. It is shown that N_{n,d} attains the branching factor ℝ_{α−β}(n)=ξ_{n}/l-ξ_{n} where ξ_{n} is the positive root of x^{n}+x-l=0. The quantity ξ_{n}/1-ξ_{n} has previously been identified as a lower bound for all directional algorithms. Thus, the equality ℝ_{α−β}(n)=ξ_{n}/1-ξ_{n} renders α-β asymptotically optimal over the class of directional, game-searching algorithms.

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## References

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1981