Independent distributions on a multi-branching AND–OR tree of height 2

Abstract

We investigate an AND–OR tree T and a probability distribution d on the truth assignments to the leaves. Tarsi (1983) inquired into an independent and identical distribution (IID) d such that at each leaf, the probability of having value 0 is neither 0 nor 1. Under a reasonable assumption on the shape of a tree, he showed the existence of an optimal algorithm that is depth-first. We investigate the case where d is an independent distribution (ID) and probability may depend on each leaf. It is known that in this general case, if the 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. This research sheds fresh light on the difference of algorithmic nature between shallow trees and deep trees.