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Multi Cut (update) - Preprint of paper Theoretical Computer...

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Preprint of paper: Theoretical Computer Science , vol. 252(1-2), pp.177–196. Multi-Cut -Pruning in Game-Tree Search Yngvi Bj¨ornsson and Tony Marsland University of Alberta, Department of Computing Science, Edmonton AB, Canada T6G 2H1 yngvi,tony @cs.ualberta.ca Abstract. The efficiency of the -algorithm as a minimax search procedure can be attributed to its effective pruning at so-called cut-nodes; ideally only one move is examined there to establish the minimax value. This paper explores the benefits of investing additional search effort at cut-nodes by also expanding some of the remaining moves. Our results show a strong correlation between the num- ber of promising move alternatives at cut-nodes and a new principal variation emerging. Furthermore, a new forward-pruning method is introduced that uses this additional information to ignore potentially futile subtrees. We also provide experimental results with the new pruning method in the domain of chess. 1 Introduction The -algorithm is the most popular method for searching game-trees in such ad- versary board games as chess, checkers and Othello. It is much more efficient than a plain brute-force minimax search because it allows a large portion of the game-tree to be pruned, while still backing up the correct game-tree value. However, the number of nodes visited by the algorithm still increases exponentially with increasing search depth. This obviously limits the scope of the search, since game-playing programs must meet external time-constraints: often having only a few minutes to make a decision. In general, the quality of play improves the further the program looks ahead 1 . Over the years, the -algorithm has been enhanced in various ways and more efficient variants have been introduced. For example, although the basic algorithm ex- plores all continuations to some fixed depth, in practice it is no longer used that way. Instead, various heuristics allow variations in the distance to the search horizon (often called the search depth or search tree height), so that some move sequences can be ex- plored more deeply than others. “Interesting” continuations are expanded beyond the nominal depth, while others are terminated prematurely. The latter case is referred to as forward-pruning , and involves some risk of overlooking a good continuation. The rationale behind the approach is that the time saved by pruning non-promising lines is better spent searching others more deeply, in an attempt to increase the overall decision quality. 1 Some artificial games have been constructed where the opposite is true; when backing up a minimax value the decision quality actually decreases with increasing search depth. This phenomenon has been studied thoroughly and is referred to as pathology in game-tree search [10]. However, such pathology is not seen in chess or the other games we are investigating.
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To effectively apply forward-pruning, good criteria are needed to determine which subtrees to ignore. Here we show that the number of good move alternatives a player has at cut-nodes can be used to identify potentially futile subtrees. Furthermore, we
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