This article has three main contributions to our understanding of minimax search: First, a new formulation for Stockman's SSS* algorithm, based on Alpha-Beta, is presented. It solves all the perceived drawbacks of SSS*, finally transforming it into a practical algorithm. In effect, we show that SSS* = Alpha-Beta + transposition tables. The crucial step is the realization that transposition tables contain so-called solution trees, structures that are used in best-first search algorithms like SSS*. Having created a practical version, we present performance measurements with tournament game-playing programs for three different mini-max games, yielding results that contradict a number of publications. Second, based on the insights gained in our attempts at understanding SSS*, we present a framework that facilitates the construction of several best-first fixed-depth game-tree search algorithms, known and new. The framework is based on depth-first null-window Alpha-Beta search, enhanced with storage to allow for the refining of previous search results. It focuses attention on the essential differences between algorithms. Third, a new instance of this framework is presented. It performs better than algorithms that are currently used in most state-of-the-art game-playing programs. We provide experimental evidence to explain why this new algorithm, MTD(ƒ), performs better than other fixed-depth minimax algorithms.
This paper proposes a new proof of Dilworth's theorem. The proof is based upon the minflow/maxcut property in flow networks. In relation to this proof, a new method to find both a Dilworth decomposition and a maximal antichain is presented.
This article has three main contributions to our understanding of minimax search:First, a new formulation for Stockman's SSS* algorithm, based on AlphaBeta, is presented. It solves all the perceived drawbacks of SSS*, finally transforming it into a practical algorithm. In effect, we show that SSS* = Alpha-Beta + transposition tables. The crucial step is the realization that transposition tables contain so-called solution trees, structures that are used in best-first search algorithms like SSS*. Having created a practical version, we present performance measurements with tournament game-playing programs for three different minimax games, yielding results that contradict a number of publications.Second, based on the insights gained in our attempts at understanding SSS*, we present a framework that facilitates the construction of several best-first fixeddepth game-tree search algorithms, known and new. The framework is based on depth-first null-window Alpha-Beta search, enhanced with storage to allow for the refining of previous search results. It focuses attention on the essential differences between algorithms.Third, a new instance of this framework is presented. It performs better than algorithms that are currently used in most state-of-the-art game-playing programs. We provide experimental evidence to explain why this new algorithm, MTD(ƒ), performs better than other fixed-depth minimax algorithms.
In this paper a theory of game tree algorithms is presented, entirely based upon the concept of a solution tree. Two t ypes of solution trees are distinguished: max and min trees. Every game tree algorithm tries to prune as many nodes as possible from the game tree. A cut-o criterion in terms of solution trees will be formulated, which can be used to eliminate nodes from the search without a ecting the result. Further, we show that any algorithm actually constructs a superposition of a max and a min solution tree. Finally, w e will see how solution trees and the related cuto criterion are applied in major game tree algorithms like alphabeta and MTD.
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