Nearly Optimal Minimax Tree Search?

TL;DR

This paper investigates the theoretical and empirical aspects of minimax search trees in game-playing, revealing that current algorithms are nearly optimal but could be improved by better understanding minimal graphs and transposition use.

Contribution

It introduces a new analysis of minimal trees and graphs in minimax search, and proposes enhancements like transposition cutoffs to improve search efficiency.

Findings

Enhanced Alpha-Beta search builds trees close to the minimal graph size.

The conventional minimal graph definition is inaccurate; real minimal graphs are smaller.

Enhanced transposition cutoffs significantly reduce search tree size.

Abstract

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.