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

  • Authors:

  • Nikos Leonardos;Michael Saks

  • Affiliations:

  • Rutgers University, Computer Science Department, 110 Frelinghuysen Rd, 08854, Piscataway, NJ, USA;Rutgers University, Mathematics Department, 110 Frelinghuysen Rd, 08854, Piscataway, NJ, USA

  • Venue:
  • Computational Complexity - Selected papers from the 24th Annual IEEE Conference on Computational Complexity (CCC 2009)
  • Year:
  • 2010

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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)/cd(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.