Pebbles and Branching Programs for Tree Evaluation

• Authors: Stephen Cook;Pierre McKenzie;Dustin Wehr;Mark Braverman;Rahul Santhanam
• Venue: ACM Transactions on Computation Theory (TOCT)
• Year: 2012

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(kh) 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(kh) 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(kh/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.