Branching programs for tree evaluation

Abstract

The problem consists FT hd (k)in computing the value in [k]={1,...,k} taken by the root of a balanced d-ary tree of height h whose internal nodes are labelled with d-ary functions on [k] and whose leaves are labelled with elements of [k]. We propose FT hd (k)as a good candidate for witnessing . We observe that the latter would follow from a proof that k-way branching programs solving FT hd (k)require ω(k unbounded function(h) size. We introduce a "state sequence" method that can match the size lower bounds on obtained by the Nečiporuk method and can yield slightly better (yet still subquadratic) bounds for some nonboolean functions. Both methods yield the tight bounds Θ(k 3) and Θ(k 5/2) for deterministic and nondeterministic branching programs solving respectively. We propose as a challenge to break the quadratic barrier inherent in the Neč iporuk method by adapting the state sequence method to handle FT 44. © 2009 Springer Berlin Heidelberg.