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

Abstract

This paper analyzes Nn,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 Nn,d attains the branching factor ℝα−β(n)=ξn/l-ξn where ξn is the positive root of xn+x-l=0. The quantity ξn/1-ξn has previously been identified as a lower bound for all directional algorithms. Thus, the equality ℝα−β(n)=ξn/1-ξn renders α-β asymptotically optimal over the class of directional, game-searching algorithms.