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) is the depth of the formula, and they are optimal up to the constant in the base of the denominator.
communication complexity read-once formulae information theory and/or trees