Lower Bounds on the Randomized Communication Complexity of Read-Once Functions

We prove lower bounds on the randomized two-party communication complexity of functions that arise from read-once Boolean formulae. A read-once Boolean formula is a formula in propositional logic with the property that every variable appears exactly once. Such a formula can be represented by a tree, where the leaves correspond to variables, and the internal nodes are labeled by binary connectives. Under certain assumptions, this representation is unique. Thus, one can define the depth of a formula as the depth of the tree that represents it. The complexity of the evaluation of general read-once formulae, has attracted interest mainly in the decision tree model. In the communication complexity model, many interesting results deal with specific read-once formulae, such as DISJOINTNESS and TRIBES. In this paper we use information theory methods to prove lower bounds that hold for any read-once formula. Our lower bounds are of the form n(f)/c^d(f), where n(f) is the number of variables and d(f) the depth of the formula, and they are optimal up to the constant c in the denominator.