The Solution for the Branching Factor of the Alpha-Beta Pruning Algorithm and its Optimality

Abstract

This paper analyzes N,d, the average number of terminal nodes examined by the a-fl 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 increasing the search depth by one extra step would increase N,d, by a factor (called the branching factor) [formula omitted] where [formula omitted] is the positive root of [formula omitted] This implies that for a given search time allotment, the [formula omitted] pruning allows the search depth to be increased by a factor [formula omitted] over that of an exhaustive minimax search. Moreover, since the quantity [formula omitted] has been identified as an absolute lower bound for the average complexity of all game searching algorithms, the equality [formula omitted] now renders [formula omitted] asymptotically optimal. © 1982, ACM. All rights reserved.