Information Equals Amortized Communication

TL;DR

This paper introduces a method to simulate message transmission with communication cost close to the information revealed, generalizing the Slepian-Wolf theorem to interactive settings and linking communication complexity to information theory.

Contribution

It generalizes the Slepian-Wolf theorem to interactive protocols and establishes a fundamental connection between amortized communication and information cost in two-party computations.

Findings

Expected communication is close to the information revealed in message simulation.

Internal information cost equals amortized communication complexity for functions.

A strong direct sum theorem hinges on efficient protocol compression.

Abstract

We show how to efficiently simulate the sending of a message M to a receiver who has partial information about the message, so that the expected number of bits communicated in the simulation is close to the amount of additional information that the message reveals to the receiver. This is a generalization and strengthening of the Slepian-Wolf theorem, which shows how to carry out such a simulation with low amortized communication in the case that M is a deterministic function of X. A caveat is that our simulation is interactive. As a consequence, we prove that the internal information cost (namely the information revealed to the parties) involved in computing any relation or function using a two party interactive protocol is exactly equal to the amortized communication complexity of computing independent copies of the same relation or function. We also show that the only way to prove a strong direct sum theorem for randomized communication complexity is by solving a particular variant of the pointer jumping problem that we define. Our work implies that a strong direct sum theorem for communication complexity holds if and only if efficient compression of communication protocols is possible.