Read-Once Branching Programs for Tree Evaluation Problems

Abstract

Toward the ultimate goal of separating L and P , Cook, McKenzie, Wehr, Braverman, and Santhanam introduced the Tree Evaluation Problem ( TEP ). For fixed integers h and k > 0, FT h ( k ) is given as a complete, rooted binary tree of height h , in which each root node is associated with a function from [ k ] 2 to [ k ], and each leaf node with a number in [ k ]. The value of an internal node v is defined naturally; that is, if it has a function f and the values of its two child nodes are a and b , then the value of v is f ( a , b ). Our task is to compute the value of the root node by sequentially executing this function evaluation in a bottom-up fashion. The problem is obviously in P , and, if we could prove that any branching program solving FT h ( k ) needs at least k r ( h ) states for any unbounded function r , then this problem is not in L , thus achieving our goal. The mentioned authors introduced a restriction called thrifty against the structure of BP’s (i,e., against the algorithm for solving the problem) and proved that any thrifty BP needs Ω( k h ) states. This article proves a similar lower bound for read-once branching programs, which allows us to get rid of the restriction on the order of nodes read by the BP that is the nature of the thrifty restriction.