Best-first fixed-depth minimax algorithms

Abstract

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.

Near-optimal control of nonlinear switched systems with non-cooperative switching rules

“…This is a minimax problem, which we solve by extending the approach from [5], called optimistic minimax search (OMS). OMS explores a tree representation of the possible sequences of max and min agent actions (here, mode switches); it is a variant of B* [17] and related to other classical minimax search methods [10], [18], [12]. It returns a near-optimal sequence with respect to the minimax-optimal value.…”

Desugaring Haskell's do-notation into applicative operations

“…• It is possible, but unimplemented, to further reduce the search space for split by observing that our dynamic programming solution has the same structure as a minimax problem allowing us to exploit alpha-beta pruning [9], computing an alpha-beta bounded transposition table rather than a classic dynamic program. MTD(f) [18] is a particularly applicable pruning technique, because the length of a do expression acts as a conservative upper bound on our cost function, our result is an integer drawn from a very small range, and our transposition table is considerably smaller than that of most games to which it has been applied. MTD(f) would not improve the worst-case cost of computing an optimal solution, but based on limited experimentation, it should bring some extreme examples, such as the one in Section 5.4, back into line with heuristic compile times.…”

Job-Level Alpha-Beta Search

“…However, since it is a BFS, search consumes a large amount of memory and therefore for problems with large sizes, its performance can often be unsatisfactory. In addition, it is difficult to demonstrate the correctness of intuitively, as pointed out by [24]. For the above problems, Plaat et al [21]- [24] proposed the MTD(f) algorithm in a BFS version, or alternatively in a depth-first search (DFS) version with the help of transposition tables to improve efficiency.…”

“…In addition, it is difficult to demonstrate the correctness of intuitively, as pointed out by [24]. For the above problems, Plaat et al [21]- [24] proposed the MTD(f) algorithm in a BFS version, or alternatively in a depth-first search (DFS) version with the help of transposition tables to improve efficiency. It obtains the benefit of expanding fewer nodes in experiments while consuming significantly less memory.…”

“…In this paper, we apply JL search to alpha-beta search [11], [19], and propose a JL alpha-beta search (JL-ABS) algorithm based on a BFS version of MTD(f), a variant of alpha-beta search proposed by Plaat et al [21], [24]. The JL-ABS algorithm is demonstrated by using it in an opening book analysis for Chinese chess.…”