On the branching factor of the alpha-beta pruning algorithm

An analysis of the alpha-beta pruning algorithm is presented which takes into account both shallow and deep cut-offs. A formula is first developed to measure the average number of terminal nodes examined by the algorithm in a uniform tree of degree n and depth d when ties are allowed among the bottom positions: specifically, all bottom values are assumed to be independent identically distributed random variables drawn from a discrete probability distribution. A worst case analysis over all possible probability distributions is then presented by considering the limiting case when the discrete probability distribution tends to a continuous probability distribution. The branching factor of the alpha-beta pruning algorithm is shown to grow with n as Θ(n/lnn), therefore confirming a claim by Knuth and Moore that deep cut-offs only have a second order effect on the behavior of the algorithm.