Independent Distributions on a Multi-Branching AND-OR Tree of Height 2

TL;DR

This paper proves that for a height-2 multi-branching AND-OR tree with independent leaf distributions, an optimal depth-first algorithm exists, extending Tarsi's result beyond the IID case.

Contribution

It establishes the existence of optimal depth-first algorithms for height-2 trees with independent, non-identical leaf distributions, a case previously unresolved.

Findings

Optimal depth-first algorithms exist for height-2 trees with independent leaf distributions.

The proof adapts Tarsi's method and uses induction on the number of leaves.

The approach does not extend to height-3 trees.

Abstract

We investigate an AND-OR tree T and a probability distribution d on the truth assignments to the leaves. Tarsi (1983) showed that if d is an independent and identical distribution (IID) such that probability of a leaf having value 0 is neither 0 nor 1 then, under a certain assumptions, there exists an optimal algorithm that is depth-first. We investigate the case where d is an independent distribution (ID) and probability depends on each leaf. It is known that in this general case, if height is greater than or equal to 3, Tarsi-type result does not hold. It is also known that for a complete binary tree of height 2, Tarsi-type result certainly holds. In this paper, we ask whether Tarsi-type result holds for an AND-OR tree of height 2. Here, a child node of the root is either an OR-gate or a leaf: The number of child nodes of an internal node is arbitrary, and depends on an internal node. We give an affirmative answer. Our strategy of the proof is to reduce the problem to the case of directional algorithms. We perform induction on the number of leaves, and modify Tarsi's method to suite height 2 trees. We discuss why our proof does not apply to height 3 trees.