Hellinger volume and number-on-the-forehead communication complexity

TL;DR

This paper introduces Hellinger volume as a new information-theoretic tool to analyze the complexity of multi-party disjointness problems in the number-on-the-forehead model, advancing understanding of communication complexity.

Contribution

It develops the concept of Hellinger volume to lower bound information cost in NOF protocols and applies it to derive new bounds for the AND_k function.

Findings

Hellinger volume lower bounds information cost in NOF protocols

New upper bound on difference between arithmetic and geometric mean

Application of tools to lower bound the complexity of AND_k

Abstract

Information-theoretic methods have proven to be a very powerful tool in communication complexity, in particular giving an elegant proof of the linear lower bound for the two-party disjointness function, and tight lower bounds on disjointness in the multi-party number-in-the-hand (NIH) model. In this paper, we study the applicability of information theoretic methods to the multi-party number-on-the-forehead model (NOF), where determining the complexity of disjointness remains an important open problem. There are two basic parts to the NIH disjointness lower bound: a direct sum theorem and a lower bound on the one-bit AND function using a beautiful connection between Hellinger distance and protocols revealed by Bar-Yossef, Jayram, Kumar and Sivakumar [BYJKS04]. Inspired by this connection, we introduce the notion of Hellinger volume. We show that it lower bounds the information cost of multi-party NOF protocols and provide a small toolbox that allows one to manipulate several Hellinger volume terms and lower bound a Hellinger volume when the distributions involved satisfy certain conditions. In doing so, we prove a new upper bound on the difference between the arithmetic mean and the geometric mean in terms of relative entropy. We then apply these new tools to obtain a lower bound on the informational complexity of the AND_k function in the NOF setting. Finally, we discuss the difficulties of proving a direct sum theorem for information cost in the NOF model.