Wim Pijls

Best-first fixed-depth minimax algorithms

Abstract This article has three main contributions to our understanding of minimax search: First, a new formulation for Stockmans 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 minimax 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( f ), performs better than other fixed-depth minimax algorithms.

A New Paradigm for Minimax Search

textabstractThis paper introduces a new paradigm for minimax game-tree search algorithms. MT is a memory-enhanced version of Pearls 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.

Complexity analysis of depth first and FP-growth implementations of APRIORI

We examine the complexity of Depth First and FP-growth implementations of Apriori, two of the fastest known data mining algorithms to find frequent itemsets in large databases. We describe the algorithms in a similar style, derive theoretical formulas, and provide experiments on both synthetic and real life data to illustrate the theory.

Another View on the SSS* Algorithm

A new version of the SSS* algorithm for searching game trees is presented. This algorithm is built around two recursive procedures. It finds the minimax value of a game tree by first establishing an upper bound to this value and then successively trying in a top down fashion to tighten this bound until the minimax value has been obtained.

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.

Game tree algorithms and solution trees

In this paper a theory of game tree algorithms is presented, entirely based upon the concept of a solution tree. Two types of solution trees are distinguished: max and min trees. Every game tree algorithm tries to prune as many nodes as possible from the game tree. A cut-off criterion in terms of solution trees will be formulated, which can be used to eliminate nodes from the search without affecting the result. Further, we show that any algorithm actually constructs a superposition of a max and a min solution tree. Finally, we will see how solution trees and the related cutoff criterion are applied in major game tree algorithms like alphabeta and MTD.

Nearly Optimal Minimax Tree Search

Knuth and Moore presented a theoretical lower bound on the number of leaves that any fixed-depth minimax tree-search algorithm traversing a uniform tree must explore, the so-called minimal tree. Since real-life minimax trees are not uniform, the exact size of this tree is not known for most applications. Further, most games have transpositions, implying that there exists a minimal graph which is smaller than the minimal tree. For three games (chess, Othello and checkers) we compute the size of the minimal tree and the minimal graph. Empirical evidence shows that in all three games, enhanced Alpha-Beta search is capable of building a tree that is close in size to that of the minimal graph. Hence, it appears game-playing programs build nearly optimal search trees. However, the conventional definition of the minimal graph is wrong. There are ways in which the size of the minimal graph can be reduced: by maximizing the number of transpositions in the search, and generating cutoffs using branches that lead to smaller search trees. The conventional definition of the minimal graph is just a left-most approximation. Calculating the size of the real minimal graph is too computationally intensive. However, upper bound approximations show it to be significantly smaller than the left-most minimal graph. Hence, it appears that game-playing programs are not searching as efficiently as is widely believed. Understanding the left-most and real minimal search graphs leads to some new ideas for enhancing Alpha-Beta search. One of them, enhanced transposition cutoffs, is shown to significantly reduce search tree size.

A Minimax Algorithm Better than Alpha-Beta? No and Yes

This paper has three main contributions to our understanding of fixed-depth minimax search: (A) A new formulation for Stockmans 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 + ransposition 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 minimax games, yielding results that contradict a number of publications. (B) 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

Game Tree Algorithms and Solution Trees

In this paper a theory of game tree algorithms is presented, entirely based upon the concept of a solution tree. Two types of solution trees are distinguished: max and min trees. Every game tree algorithm tries to prune as many nodes as possible from the game tree. A cut-off criterion in terms of solution trees will be formulated, which can be used to eliminate nodes from the search without affecting the result. Further, we show that any algorithm actually constructs a superposition of a max and a min solution tree. Finally, we will see how solution trees and the related cutoff criterion are applied in major game tree algorithms like alphabeta and MTD.

A Theory of Game Trees, Based on Solution Trees

In this paper, a theory of game tree algorithms is presented, entirely based upon the concept of solution tree. During execution of a game tree algorithm, one may distinguish between so-called alive and dead nodes. It will turn out, that only alive nodes have to be considered, whereas dead nodes should be neglected. The algorithm may stop, when every node is dead. Further, it is proved that every algorithm needs to build a critical tree. Finally, we show, that some common game tree algorithms agree with this theory.