Non-Depth-First Search against Independent Distributions on an AND-OR Tree

TL;DR

This paper investigates the optimal algorithms for AND-OR trees under independent distributions, showing that non-depth-first algorithms can be optimal and extending previous results to more general cases.

Contribution

It proves that non-depth-first algorithms can be optimal for AND-OR trees under independent distributions, extending prior work to multi-branching trees and general distributions.

Findings

Non-depth-first algorithms can be optimal for certain distributions.

Extension of previous results to multi-branching trees.

Optimal algorithms can be chosen from depth-first algorithms under equilibrium conditions.

Abstract

Suzuki and Niida (Ann. Pure. Appl. Logic, 2015) showed the following results on independent distributions (IDs) on an AND-OR tree, where they took only depth-first algorithms into consideration. (1) Among IDs such that probability of the root having value 0 is fixed as a given r such that 0 < r < 1, if d is a maximizer of cost of the best algorithm then d is an independent and identical distribution (IID). (2) Among all IDs, if d is a maximizer of cost of the best algorithm then d is an IID. In the case where non-depth-first algorithms are taken into consideration, the counter parts of (1) and (2) are left open in the above work. Peng et al. (Inform. Process. Lett., 2017) extended (1) and (2) to multi-branching trees, where in (2) they put an additional hypothesis on IDs that probability of the root having value 0 is neither 0 nor 1. We give positive answers for the two questions of Suzuki-Niida. A key to the proof is that if ID d achieves the equilibrium among IDs then we can chose an algorithm of the best cost against d from depth-first algorithms. In addition, we extend the result of Peng et al. to the case where non-depth-first algorithms are taken into consideration.