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<p>Investigation of several algorithms for computing exact minimax values of game trees (utilizing backward pruning) are discussed. The focus is on trees with an ordering similar to that actually found in game playing practice. The authors compare the algorithms using two different distributions of the static values, the uniform distribution and a distribution estimated from practical data. A systematic comparison of using aspiration windows for all of the usual minimax algorithms is presented. The effects of aspiration windows of varying size and position are analyzed. Increasing the ordering of moves to near the optimum results in unexpectedly high savings. Algorithms with linear space complexity benefit most. Although the ordering of the first move is of predominant importance, that of the remainder has only second-order effects. The use of an aspiration window not only makes alpha-beta search competitive, but there also exist dependencies of its effects on certain properties of the trees.</p>
minimax search algorithms; aspiration windows; exact minimax values; game trees; backward pruning; uniform distribution; linear space complexity; alpha-beta search; computational complexity; game theory; minimax techniques; search problems; trees (mathematics)

R. Shams, H. Horacek and H. Kaindl, "Minimax Search Algorithms With and Without Aspiration Windows," in IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 13, no. , pp. 1225-1235, 1991.
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