Communication Complexity of Collision

TL;DR

This paper establishes a polynomial lower bound on the randomized communication complexity of the Collision problem, revealing fundamental limits on distributed decision-making for this problem.

Contribution

It proves a new polynomial lower bound for the two-party Collision problem's randomized communication complexity, addressing an open question in the field.

Findings

Polynomial lower bound for Collision problem

Lower bound for weak bit-pigeonhole search problem

Answers open question by Itsykson and Riazanov

Abstract

The Collision problem is to decide whether a given list of numbers $(x_1,\ldots,x_n)\in[n]^n$ is $1$-to-$1$ or $2$-to-$1$ when promised one of them is the case. We show an $n^{\Omega(1)}$ randomised communication lower bound for the natural two-party version of Collision where Alice holds the first half of the bits of each $x_i$ and Bob holds the second half. As an application, we also show a similar lower bound for a weak bit-pigeonhole search problem, which answers a question of Itsykson and Riazanov (CCC 2021).