Separation and Collapse of Equilibria Inequalities on AND-OR Trees without Shape Constraints

TL;DR

This paper analyzes the complexity of AND-OR tree evaluation with randomized algorithms, showing that depth-first algorithms match directional algorithms in equilibrium, and introduces a new algorithm to demonstrate separation in equilibria inequalities.

Contribution

It proves the collapse of equilibria inequalities for all AND-OR trees and introduces a new algorithm to establish separation results.

Findings

Depth-first algorithms share equilibrium with directional algorithms.

Collapse of equilibria inequalities applies to arbitrary AND-OR trees.

A new algorithm demonstrates the separation in equilibria inequalities.

Abstract

Herein, we investigate the zero-error randomized complexity, which is the least cost against the worst input, of AND-OR tree computation by imposing various restrictions on the algorithm to find the Boolean value of the root of that tree and no restrictions on the tree shape. When a tree satisfies a certain condition regarding its symmetry, directional algorithms proposed by Saks and Wigderson (1986), special randomized algorithms, are known to achieve the randomized complexity. Furthermore, there is a known example of a tree that is so unbalanced that no directional algorithm achieves the randomized complexity (Vereshchagin 1998). In this study, we aim to identify where deviations arise between the general randomized Boolean decision tree and its special case, directional algorithms. We show that for any AND-OR tree, randomized depth-first algorithms, which form a broader class compared with directional algorithms, have the same equilibrium as that of the directional algorithms. Thus, we get the collapse result on equilibria inequalities that holds for an arbitrary AND-OR tree. This implies that there exists a case where even depth-first algorithms cannot be the fastest, leading to the separation result on equilibria inequality. Additionally, a new algorithm is introduced as a key concept for proof of the separation result.