On the Role of Shared Randomness in Simultaneous Communication

Abstract

Suppose two parties who are interested in performing certain distributed computational tasks are given access to a source of correlated random bits ρ. This source of correlated randomness could be quite useful to the parties for solving various distributed computational problems as it enables the parties to act in a correlated manner. In this work, we initiate the study of power of different sources of shared randomness ρ in the setting of communication complexity; we shall do so in the model of simultaneous message passing (SMP) model of communication complexity, and we shall also argue that this model is the appropriate choice among the commonly studied models of two-party communication complexity for the purpose of studying shared randomness as a resource. As such, we introduce a natural measure for the strength of the correlation provided by a bipartite distribution that we call collision complexity. We demonstrate that the collision complexity col _ρ (n) of a bipartite distribution ρ tightly characterises the power of ρ as a resource. We also uncover some surprising phenomenon by showing that even the noisiest shared randomness increases the power of SMP substantially: the equality function can be solved very efficiently with virtually any nontrivial shared randomness— whereas without shared randomness the complexity is known to be $\Omega(\sqrt n)$.