Search

We describe a very simple selective search algorithm for two-player games, called best-first minimax. It always expands next the node at the end of the expected line of play, which determines the minimax value of the root. It uses the same information as alpha-beta minimax, and takes roughly the same time per node generation. We present an implementation of the algorithm that reduces its space complexity from exponential to linear in the search depth, but at significant time cost. Our actual implementation saves the subtree generated for one move that is still relevant after the player and opponent move, pruning subtrees below moves not chosen by either player. We also show how to efficiently generate a class of incremental random game trees. On uniform random game trees, best-first minimax outperforms alpha-beta, when both algorithms are given the same amount of computation. On random trees with random branching factors, best-first outperforms alpha-beta for shallow depths, but eventually loses at greater depths. We obtain similar results in the game of Othello. Finally, we present a hybrid best-first extension algorithm that combines alpha-beta with best-first minimax, and performs significantly better than alpha-beta in both domains, even at greater depths. In Othello, it beats alpha-beta in two out of three games.

Korf, R.E., Multi-player alpha-beta pruning (Research Note), Artificial Intelligence 48 (1991) 99-111. We consider the generalization of minimax search with alpha-beta pruning to non-cooperative, perfect-information games with more than two players. The minimax algorithm was generalized in [2] to the maxn algorithm applied to vectors of n-tuples representing the evaluations for each of the players. If we assume an upper bound on the sum of the evaluations for each player, and a lower bound on each individual evaluation, then shallow alpha-beta pruning is possible, but not deep pruning. In the best case, the asymptotic branching factor is reduced to (1 + 4bv'~b-"s~-3)/2. In the average case, however, pruning does not reduce the asymptotic branching factor. Thus, alpha-beta pruning is found to be effective only in the special case of two-player games. In addition, we show that it is an optimal directional algorithm for two players.

The alpha-beta technique for searching game trees is analyzed, in an attempt to provide some insight into its behavior. The first portion of this paper is an expository presentation of the method together with a proof of its correctness and a historical discussion. The alpha-beta procedure is shown to be optimal in a certain sense, and bounds are obtained for its running time with various kinds of random data.

Pruning decision trees is a useful technique for improving the generalization performance in decision tree induction, and for trading accuracy for simplicity in other applications. In this paper, a new algorithm called OPT-2 for optimal pruning of decision trees is introduced. The algorithm is based on dynamic programming. In its most basic form, the time and space complexities of OPT-2 are both Θ (nC), where n is the number of test nodes in the initial decision tree, and C is the number of leaves in the target (pruned) decision tree. This is an improvement over the recently published OPT algorithm of Bohanec and Bratko (which is the only known algorithm for optimal decision tree pruning) especially in the case of heavy pruning and when the tests of the given decision tree have many outcomes. If so desired, the space required by OPT-2 can further be reduced by a factor of r at the cost of increasing the execution time by a factor that is bounded above by (r + 1) 2 (this is a considerable overestimate, however). From a practical point of view, OPT-2 enjoys considerable flexibility in various aspects, and is easy to implement.