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Solution Trees as a Basis for Game Tree Search Arie de Bruin, Wim Pijls, Aske Plaat Erasmus University, Department of Computer Science P.O.Box 1738, 3000 DR Rotterdam, The Netherlands [email protected] September 20, 1994 Abstract A game tree algorithm is an algorithm computing the minimax value of the root of a game tree. Two well-known game tree search algorithms are alpha-beta and SSS*. We show a relation between these two algorithms, that are commonly regarded as being quite different. Many algorithms use the notion of establishing proofs that the game value lies above or below some boundary value. We show that this amounts to the construction of a solution tree. We discuss the role of solution trees and critical trees [KM75] in the following algorithms: Principal Variation Search, alpha-beta, and SSS*. A general procedure for the construction of a solution tree, based on alpha-beta and Null-Window- Search, is given. Furthermore three new examples of solution tree based-algorithms are presented, which surpass alpha-beta—i.e., never visit more nodes than alpha-beta, and often less. Keywords: Game tree search, alpha-beta, SSS*, solution trees. 1 Introduction In the field of game tree search the alpha-beta algorithm has been in use since the 1950’s. It has proven quite successful, mainly due to the good results that have been achieved by programs that use it. No other algorithm has achieved the wide-spread use in practical applications that alpha-beta has. This does not mean that alpha-beta is the only algorithm for game tree search. Over the years a number of alternatives have been published. Among these are minimal win- dow algorithms like PVS [CM83, Pea84], Proof-Number Search [AvdMvdH94], Best-First Minimax Search [Kor93], and SSS* [Sto79]. The last one, SSS*, has sparked quite some research activity. This may have been caused in part by the slightly provocative nature of the title of Stockman’s original paper: “A Minimax Algorithm Better than Alpha-Beta?”. This title alone has provoked a few reactions in the form of papers by Roizen and Pearl (“Yes and No” [RP83]), and Reinefeld (“A Minimax Algorithm Faster than Alpha-Beta” [Rei94]). In the present paper we investigate the relation between alpha-beta, PVS, and SSS*. We confine ourselves to the basic algorithms, without enhancements like move-reordering, iterative deepening, or transposition tables (see e.g. [CM83, Sch89, ACH90]). Alpha-beta, being a strictly depth-first algorithm, is generally regarded to be quite different in nature from best-first algorithms like SSS*. We will try to show in this paper how these algorithms are related. This paper is also registered as Technical Report EUR-CS-94-04 Tinbergen Institute, Erasmus University, and Department of Computer Science, Erasmus University.
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