Solution trees as a basis for game tree search

A game tree algorithm is an algorithm computing the minimax value of the root of a game tree. Many algorithms use the notion of establishing proofs that this 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 in the following algorithms: Principal Variation Search, alpha-beta, and SSS-2. A general procedure for the construction of a solution tree, based on alpha-beta and Null-Window-Search, is given. Furthermore two new examples of solution tree-based algorithms are presented, that surpass alpha-beta, i.e., never visit more nodes than alpha-beta, and often less

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

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. This approach has several advantages, most notably that the algorithm is more perspicuous. Correctness and several other properties of SSS* can now more easily be proven. As an example we prove Pearl's characterization of the nodes visited by SSS*. Finally the new algorithm is transformed into a practical version, which allows an efficient use of memory

A theory of game trees, based on solution trees

In this paper a complete theory of game tree algorithms is presented, entirely based upon the notion of a solution tree. Two types of solution trees are distinguished: max and min solution trees respectively. We show that most game tree algorithms construct a superposition of a max and a min solution tree. Moreover, we formulate a general cut-off criterion in terms of solution trees. In the second half of this paper four well known algorithms, viz., alphabeta, SSS*, MTD and Scout are studied extensively. We show how solution trees feature in these algorithms and how the cut-off criterion is applied

The key node method: a highly-parallel alpha-beta algorithm

Journal ArticleA new parallel formulation of the alpha-beta algorithm for minimax game tree searching is presented. Its chief characteristic is incremental information sharing among subsearch processes in the form of "provisional" node value communication. Such "eager" communication can offer the double benefit of faster search focusing and enhanced parallelism. This effect is particularly advantageous in the prevalent case when static value correlation exists among adjacent nodes. A message-passing formulation of this idea, termed the "Key Node Method", is outlined. Preliminary experimental results for this method are reported, supporting its validity and potential for increased speedup

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