Communication Complexity of Disjointness under Product Distributions

TL;DR

This paper revisits the communication complexity of the disjointness problem under product distributions, providing a simplified proof of the best known bounds with improved error dependence, using a novel combinatorial lemma.

Contribution

It offers a streamlined proof of tight bounds for disjointness communication complexity under product distributions, introducing a new combinatorial lemma of independent interest.

Findings

Established tight bounds for disjointness under product distributions.

Provided a simplified proof with better error dependence.

Introduced a combinatorial lemma for disjointness analysis.

Abstract

Determining the randomized (or distributional) communication complexity of disjointness is a central problem in communication complexity, having roots in the foundational work of Babai, Frankl, and Simon in the 1980s and culminating in the famous works of Kalyanasundaram-Schnitger and Razborov in 1992. However, the question of obtaining tight bounds for product distributions persisted until the more recent work of Bottesch, Gavinsky, and Klauck resolved it. In this note we revisit this classical problem and give a short, streamlined proof of the best bounds, with improved quantitative dependence on the error parameter. Our approach is based on a simple combinatorial lemma that may be of independent interest: if two sets drawn independently from two distributions are disjoint with non-negligible probability, then one can extract two subfamilies of reasonably large measure that are fully cross-disjoint (equivalently, a large monochromatic rectangle for disjointness).