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TR09-010 | 29th January 2009 00:00
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#### Lower bounds on the randomized communication complexity of read-once functions

**Abstract:**
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.