A new paradigm for minimax search

This paper introduces a new paradigm for minimax game-tree search algorithms. MT is a memory-enhanced version of Pearl's Test procedure. By changing the way MT is called, a number of best-first game-tree search algorithms can be simply and elegantly constructed (including SSS*). Most of the assessments of minimax search algorithms have been based on simulations. However, these simulations generally do not address two of the key ingredients of high performance game-playing programs: iterative deepening and memory usage. This paper presents experimental data from three game-playing programs (checkers, Othello and chess), covering the range from low to high branching factor. The improved move ordering due to iterative deepening and memory usage results in significantly different results from those portrayed in the literature. Whereas some simulations show alpha-beta expanding almost 100% more leaf nodes than other algorithms [Marsland, Reinefeld & Schaeffer, 1987], our results showed variations of less than 20%. One new instance of our framework MTD(f) out-performs our best alpha-beta searcher (aspiration NegaScout) on leaf nodes, total nodes and execution time. To our knowledge, these are the first reported results that compare both depth-first and best-first algorithms given the same amount of memory

Trends in game tree search

This paper deals with algorithms searching trees generated by two-person, zero-sum games with perfect information. The standard algorithm in this field is alpha-beta. We will discuss this algorithm as well as extensions, like transposition tables, iterative deepening and NegaScout. Special attention is devoted to domain knowledge pertaining to game trees, more specifically to solution trees. The above mentioned algorithms implement depth first search. The alternative is best first search. The best known algorithm in this area is Stockman's SSS*. We treat a variant equivalent to SSS* called SSS-2. These algorithms are provably better than alpha-beta, but it needs a lot of tweaking to show this in practice. A variant of SSS-2, cast in alpha-beta terms, will be discussed which does realize this potential. This algorithm is however still worse than NegaScout. On the other hand, applying a similar idea as the one behind NegaScout to this last SSS version yields the best (sequential) game tree searcher known up till now: MTD(f)

The Analysis of Alpha Beta Pruning and MTD(f) Algorithm to Determine the Best Algorithm to be Implemented at Connect Four Prototype

Connect Four is a two-player game which the players take turns dropping discs into a grid to connect 4 of one’s own discs next to each other vertically, horizontally, or diagonally. At Connect Four, Computer requires artificial intelligence (AI) in order to play properly like human. There are many AI algorithms that can be implemented to Connect Four, but the suitable algorithms are unknown. The suitable algorithm means optimal in choosing move and its execution time is not slow at search depth which is deep enough. In this research, analysis and comparison between standard alpha beta (AB) Pruning and MTD(f) will be carried out at the prototype of Connect Four in terms of optimality (win percentage) and speed (execution time and the number of leaf nodes). Experiments are carried out by running computer versus computer mode with 12 different conditions, i.e. varied search depth (5 through 10) and who moves first. The percentage achieved by MTD(f) based on experiments is win 45,83%, lose 37,5% and draw 16,67%. In the experiments with search depth 8, MTD(f) execution time is 35, 19% faster and evaluate 56,27% fewer leaf nodes than AB Pruning. The results of this research are MTD(f) is as optimal as AB Pruning at Connect Four prototype, but MTD(f) on average is faster and evaluates fewer leaf nodes than AB Pruning. The execution time of MTD(f) is not slow and much faster than AB Pruning at search depth which is deep enough

Best-First and Depth-First Minimax Search in practice

Abstract Most practitioners use a variant of the Alpha-Beta algorithm, a simple depth-first procedure, for searching minimax trees. SSS*, with its best-first search strategy, reportedly offers the potential for more efficient search. However, the complex formulation of the algorithm and its alleged excessive memory requirements preclude its use in practice. For two decades, the search efficiency of "smart" best-first SSS* has cast doubt on the effectiveness of "dumb" depth-first Alpha-Beta

Intelligent strategy for two-person non-random perfect information zero-sum game.

Tong Kwong-Bun.Thesis submitted in: December 2002.Thesis (M.Phil.)--Chinese University of Hong Kong, 2003.Includes bibliographical references (leaves 77-[80]).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- An Overview --- p.1Chapter 1.2 --- Tree Search --- p.2Chapter 1.2.1 --- Minimax Algorithm --- p.2Chapter 1.2.2 --- The Alpha-Beta Algorithm --- p.4Chapter 1.2.3 --- Alpha-Beta Enhancements --- p.5Chapter 1.2.4 --- Selective Search --- p.9Chapter 1.3 --- Construction of Evaluation Function --- p.16Chapter 1.4 --- Contribution of the Thesis --- p.17Chapter 1.5 --- Structure of the Thesis --- p.19Chapter 2 --- The Probabilistic Forward Pruning Framework --- p.20Chapter 2.1 --- Introduction --- p.20Chapter 2.2 --- The Generalized Probabilistic Forward Cuts Heuristic --- p.21Chapter 2.3 --- The GPC Framework --- p.24Chapter 2.3.1 --- The Alpha-Beta Algorithm --- p.24Chapter 2.3.2 --- The NegaScout Algorithm --- p.25Chapter 2.3.3 --- The Memory-enhanced Test Algorithm --- p.27Chapter 2.4 --- Summary --- p.27Chapter 3 --- The Fast Probabilistic Forward Pruning Framework --- p.30Chapter 3.1 --- Introduction --- p.30Chapter 3.2 --- The Fast GPC Heuristic --- p.30Chapter 3.2.1 --- The Alpha-Beta algorithm --- p.32Chapter 3.2.2 --- The NegaScout algorithm --- p.32Chapter 3.2.3 --- The Memory-enhanced Test algorithm --- p.35Chapter 3.3 --- Performance Evaluation --- p.35Chapter 3.3.1 --- Determination of the Parameters --- p.35Chapter 3.3.2 --- Result of Experiments --- p.38Chapter 3.4 --- Summary --- p.42Chapter 4 --- The Node-Cutting Heuristic --- p.43Chapter 4.1 --- Introduction --- p.43Chapter 4.2 --- Move Ordering --- p.43Chapter 4.2.1 --- Quality of Move Ordering --- p.44Chapter 4.3 --- Node-Cutting Heuristic --- p.46Chapter 4.4 --- Performance Evaluation --- p.48Chapter 4.4.1 --- Determination of the Parameters --- p.48Chapter 4.4.2 --- Result of Experiments --- p.50Chapter 4.5 --- Summary --- p.55Chapter 5 --- The Integrated Strategy --- p.56Chapter 5.1 --- Introduction --- p.56Chapter 5.2 --- "Combination of GPC, FGPC and Node-Cutting Heuristic" --- p.56Chapter 5.3 --- Performance Evaluation --- p.58Chapter 5.4 --- Summary --- p.63Chapter 6 --- Conclusions and Future Works --- p.64Chapter 6.1 --- Conclusions --- p.64Chapter 6.2 --- Future Works --- p.65Chapter A --- Examples --- p.67Chapter B --- The Rules of Chinese Checkers --- p.73Chapter C --- Application to Chinese Checkers --- p.75Bibliography --- p.7