Read-Once Branching Programs for Tree Evaluation Problems

Toward the ultimate goal of separating L and P, Cook, McKenzie, Wehr, Braverman and Santhanam introduced the tree evaluation problem (TEP). For fixed h, k>0, FT_h(k) is given as a complete, rooted binary tree of height h, in which each internal 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, i.e., 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 above 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 Omega(k^h) states. This paper 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.