Tight bounds on the randomized communication complexity of symmetric XOR functions in one-way and SMP models

TL;DR

This paper establishes tight bounds on the randomized communication complexity of symmetric XOR functions in one-way and SMP models, closing a significant gap between quantum lower bounds and randomized upper bounds.

Contribution

It provides a public-coin randomized protocol in the SMP model matching quantum and two-way lower bounds up to a logarithmic factor, resolving an open problem.

Findings

Communication cost matches quantum and two-way lower bounds up to a log factor

Closes quadratic gap between quantum lower bounds and randomized upper bounds in one-way model

Answers an open question from Shi and Zhang (2009)

Abstract

We study the communication complexity of symmetric XOR functions, namely functions $f: \{0,1\}^n \times \{0,1\}^n \rightarrow \{0,1\}$ that can be formulated as $f(x,y)=D(|x\oplus y|)$ for some predicate $D: \{0,1,...,n\} \rightarrow \{0,1\}$, where $|x\oplus y|$ is the Hamming weight of the bitwise XOR of $x$ and $y$. We give a public-coin randomized protocol in the Simultaneous Message Passing (SMP) model, with the communication cost matching the known lower bound for the \emph{quantum} and \emph{two-way} model up to a logarithm factor. As a corollary, this closes a quadratic gap between quantum lower bound and randomized upper bound for the one-way model, answering an open question raised in Shi and Zhang \cite{SZ09}.