How to compress interactive communication

Abstract

We describe new ways to simulate 2-party communication protocols to get protocols with potentially smaller communication. We show that every communication protocol that communicates C bits and reveals I bits of information about the inputs to the participating parties can be simulated by a new protocol involving at mostÕ( √ CI) bits of communication. If the protocol reveals I bits of information about the inputs to an observer that watches the communication in the protocol, we show how to carry out the simulation withÕ(I) bits of communication.These results lead to a direct sum theorem for randomized communication complexity. Ignoring polylogarithmic factors, we show that for worst case computation, computing n copies of a function requires √ n times the communication required for computing one copy of the function. For average case complexity, given any distribution µ on inputs, computing n copies of the function on n inputs sampled independently according to µ requires √ n times the communication for computing one copy. If µ is a product distribution, computing n copies on n independent inputs sampled according to µ requires n times the communication required for computing the function. We also study the complexity of computing the sum (or parity) of n evaluations of f , and obtain results analogous to those above.To the best of our knowledge, our results give the first compression schemes for general randomized protocols and the first direct sum results in the general setting. Previous results applied only when the protocols were restricted to running in a bounded number of rounds, where each message can be compressed in turn, and only applied when the parties are given independent inputs.

Direct Products in Communication Complexity

“…Direct sum theorems for this model were proved in [BBCR10], and we strengthen their results to give direct product theorems. For a longer introduction to direct sums and direct products in communication complexity and their significance, we refer the reader to the introductions of [BBCR10,JPY12].…”

“…Our proofs heavily rely on methods from information theory [Sha48] which have been applied to a variety of problems in communication complexity [Raz92,NW93,Abl96,CSWY01,BYJKS04,BBCR10], and ideas developed to prove the parallel repetition theorem [Raz98,Hol07]. We give an overview of our proofs next.…”

“…Before we describe our proof in detail, we give a high level overview of the proof of the direct sum theorem proved in [BBCR10]. The theorem is proved by reduction.…”

“…The bound is meaningful only when n > C and the protocol for n copies is not allowed to communicate more bits than the protocol for 1 copy. We refer the reader to [BBCR10,JPY12] for more references.…”

“…Prior to our work, the only known general upper bounds on suc(µ n , f n , T ), for T > C, are a consequence of the direct sum theorem proved in [BBCR10]: If suc(µ, f, C) ≤ 2 3 , then suc(µ n , f n , T ) ≤ 2 3 , as long as T log T C √ n. They also proved the same upper bound when T polylog(T ) Cn and µ is a product distribution.…”