Experiments With Some Programs That Search Game Trees
Journal of the ACM (JACM)
A quantitative analysis of the alpha-beta pruning algorithm
Artificial Intelligence
A minimax algorithm better than alpha-beta? Yes and No
Artificial Intelligence
Pathology on game trees revisited, and an alternative to minimaxing
Artificial Intelligence
The *-minimax search procedure for trees containing chance nodes
Artificial Intelligence
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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 @Q(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.