Communication Complexity of Disjointness under Product Distributions

2026-03-19

AI Summary

This work investigates the randomized communication complexity of the disjointness problem (DISJ) under product distributions, with a focus on obtaining a tight characterization of its dependence on the error parameter. The authors introduce a novel and concise combinatorial lemma stating that if two independent distributions generate disjoint sets with non-negligible probability, then one can extract large-measure subfamilies that are completely disjoint. Leveraging this lemma alongside probabilistic methods and rectangle decomposition techniques, they significantly refine the quantitative dependence of the communication complexity bound on the error probability. This yields the current best-known tight bound, thereby advancing the precise understanding of how communication complexity scales with the allowed error in the product distribution setting.

Technology Category

Constraint Satisfaction and Optimization: Distributed CSP/OptimizationSearch and Optimization: Distributed SearchReasoning under Uncertainty: Stochastic Optimization

Application Category

Security and Privacy: Large-scale security measurementsGraph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphsEconomics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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).

Problem

• communication complexity
• disjointness
• product distributions
• randomized protocols
• distributional complexity

Innovation

• communication complexity
• disjointness
• product distributions
• combinatorial lemma
• monochromatic rectangle