RESEARCH RE: SEARCH & RE-SEARCH

Search algorithms are often categorized by their node expansion strategy. One option is the depth-first strategy, a simple backtracking strategy that traverses the search space in the order in which successor nodes are generated. An alternative is the best-first strategy, which was designed to make it possible to use domain-specific heuristic information. By exploring promising parts of the search space first, best-first algorithms are usually more efficient than depth-first algorithms. In programs that play minimax games such as chess and checkers, the efficiency of the search is of crucial importance. Given the success of best-first algorithms in other domains, one would expect them to be used for minimax games too. However, all high-performance game-playing programs are based on a depth-first algorithm. This study takes a closer look at a depth-first algorithm, Alpha-Beta, and a best-first algorithm, SSS*. The prevailing opinion on these algorithms is that SSS* offers the potential for a more efficient search, but that its complicated formulation and exponential memory requirements render it impractical. The theoretical part of this work shows that there is a surprisingly straightforward link between the two algorithms—for all practical purposes, SSS* is a special case of Alpha-Beta. Subsequent empirical evidence proves the prevailing opinion on SSS* to be wrong: it is not a complicated algorithm, it does not need too much memory, and it is also not more efficient than depth-first search. Over the years, research on Alpha-Beta has yielded many enhancements, such as transposition tables and minimal windows with re-searches, that are responsible for the success of depth-first minimax search. The enhancements have made it possible to use a depth-first procedure to expand nodes in a best-first sequence. Based on these insights, a new algorithm is presented, MTD(ƒ), which out-performs both SSS* and NegaScout, the Alpha-Beta variant of choice by practitioners. In addition to best-first search, other ways for improvement of minimax search algorithms are investigated. The tree searched in Alpha-Beta’s best case is usually considered to be equal to the minimal tree that has to be searched by any algorithm in order to find and prove the minimax value. We show that in practice this assumption is not valid. For non-uniform trees, the real minimal tree—or rather, graph—that proves the minimax value is shown to be significantly smaller than Alpha-Beta’s best case. Thus, there is more room for improvement of full-width minimax search than is generally assumed.