Pebbling and Branching Programs Solving the Tree Evaluation Problem

We study restricted computation models related to the tree evaluation problem. The TEP was introduced in earlier work as a simple candidate for the (very) long term goal of separating L and LogDCFL. The input to the problem is a rooted, balanced binary tree of height h, whose internal nodes are labeled with binary functions on [k] = {1, . . . , k} (each given simply as a list of k2 elements of [k]), and whose leaves are labeled with elements of [k]. Each node obtains a value in [k] equal to its binary function applied to the values of its children. The output is the value of the root. The first restricted computation model, called fractional pebbling, is a generalization of the black/white pebbling game on graphs, and arises in a natural way from the search for good upper bounds on the size of nondeterministic branching programs solving the TEP for any fixed h, if the binary tree of height h has fractional pebbling cost at most p, then there are nondeterministic branching programs of size O(kp) solving the height h TEP. We prove a lower bound on the fractional pebbling cost of d-ary trees that is tight to within an additive constant for each fixed d. The second restricted computation model we study is a semantic restriction on (non)deterministic branching programs solving the TEP – thrifty branching programs. Deterministic (resp. nondeterministic) thrifty BPs suffice to implement the best known algorithms, based on black pebbling (resp. fractional pebbling), for the TEP. In earlier work, for each fixed h a lower bound on the size of thrifty deterministic branching programs was proved that is tight for sufficiently large k. We give an alternative proof that achieves the same bound for all k and h. We show the same bound still holds in a lessrestricted model, and also that gradually weaker lower bounds can be obtained for gradually weaker restrictions on the model.