New approaches for understanding the asymptotic complexity of A^* tree searching

Abstract

Previous studies of A^* tree-searching have modeled heuristics as random variables. The average number of nodes expanded is expressed asymptotically in terms of distance to goal. The conclusion reached is that A^* complexity is an exponential function of heuristic error: Polynomial error implies exponential complexity and logarithmic accuracy is required for polynomial complexity.

This paper eliminates simplifying assumptions of earlier studies. “Error” is replaced by a concept called “discrepancy”, a measure of the relative attractiveness to A^* of a node for expansion when that node is compared with competing nodes on the solution path. According to our model, in order to have polynomial A^* complexity, it is not necessary to have the “logarithmic accuracy” described in previous studies. Another way is for a heuristic's values to vary, or cluster, near a “central function” which grows at least as fast as distance to goal. Generally, logarithmic variation or less is adequate. For one class of heuristics considered, the faster this central function grows, the more is variation from it tolerated.