Pebbles and Branching Programs for Tree Evaluation

Abstract

We introduce the tree evaluation problem , show that it is in LogDCFL (and hence in P ), and study its branching program complexity in the hope of eventually proving a superlogarithmic space lower bound. The input to the problem is a rooted, balanced d -ary tree of height h , whose internal nodes are labeled with d -ary functions on [ k ] = {1,..., k }, and whose leaves are labeled with elements of [ k ]. Each node obtains a value in [ k ] equal to its d -ary function applied to the values of its d children. The output is the value of the root. We show that the standard black pebbling algorithm applied to the binary tree of height h yields a deterministic k -way branching program with O ( k h ) states solving this problem, and we prove that this upper bound is tight for h  = 2 and h  = 3. We introduce a simple semantic restriction called thrifty on k -way branching programs solving tree evaluation problems and show that the same state bound of Θ ( k h ) is tight for all h  ≥ 2 for deterministic thrifty programs. We introduce fractional pebbling for trees and show that this yields nondeterministic thrifty programs with Θ ( k h/2+1 ) states solving the Boolean problem “determine whether the root has value 1”, and prove that this bound is tight for h  = 2,3,4. We also prove that this same bound is tight for unrestricted nondeterministic k -way branching programs solving the Boolean problem for h  = 2,3.