Communication with Imperfectly Shared Randomness

TL;DR

This paper investigates how shared correlations, which are not perfectly shared randomness, affect communication complexity in interactive protocols, establishing bounds and demonstrating the tightness of these bounds.

Contribution

It provides the first general bounds relating imperfectly shared randomness to perfectly shared randomness in interactive communication complexity.

Findings

Protocols with imperfectly shared randomness require exponential overhead compared to perfect sharing.

The exponential gap between perfect and imperfect sharing is tight, demonstrated by a specific promise problem.

The work applies advanced mathematical tools like small-set expansion and invariance principles to prove bounds.

Abstract

The communication complexity of many fundamental problems reduces greatly when the communicating parties share randomness that is independent of the inputs to the communication task. Natural communication processes (say between humans) however often involve large amounts of shared correlations among the communicating players, but rarely allow for perfect sharing of randomness. Can the communication complexity benefit from shared correlations as well as it does from shared randomness? This question was considered mainly in the context of simultaneous communication by Bavarian et al. (ICALP 2014). In this work we study this problem in the standard interactive setting and give some general results. In particular, we show that every problem with communication complexity of $k$ bits with perfectly shared randomness has a protocol using imperfectly shared randomness with complexity $\exp(k)$ bits. We also show that this is best possible by exhibiting a promise problem with complexity $k$ bits with perfectly shared randomness which requires $\exp(k)$ bits when the randomness is imperfectly shared. Along the way we also highlight some other basic problems such as compression, and agreement distillation, where shared randomness plays a central role and analyze the complexity of these problems in the imperfectly shared randomness model. The technical highlight of this work is the lower bound that goes into the result showing the tightness of our general connection. This result builds on the intuition that communication with imperfectly shared randomness needs to be less sensitive to its random inputs than communication with perfectly shared randomness. The formal proof invokes results about the small-set expansion of the noisy hypercube and an invariance principle to convert this intuition to a proof, thus giving a new application domain for these fundamental results.