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An XOR Lemma for Deterministic Communication Complexity

Siddharth Iyer, Anup Rao

TL;DR

The paper establishes a lower bound on the deterministic communication complexity of computing the parity $f^{\oplus n}$ in terms of the original complexity $D(f)$ and the rank $\mathsf{rk}(f)$: $D(f^{\oplus n}) \ge n \big( \frac{\Omega(D(f))}{\log \mathsf{rk}(f)} - \log \mathsf{rk}(f) \big)$. A central contribution is a novel information-theoretic method for deterministic protocols that avoids introducing errors, together with a subadditivity lemma: if $f^{\oplus n}$ has a monochromatic rectangle of size $2^k$, then $f$ has a rectangle of size at least $2^{k/n-2}$. This connection between rectangle structure at different levels, combined with a rank-based recursive decomposition, yields the main bound and advances understanding of how self-composition impacts deterministic communication complexity.

Abstract

We prove a lower bound on the communication complexity of computing the $n$-fold xor of an arbitrary function $f$, in terms of the communication complexity and rank of $f$. We prove that $D(f^{\oplus n}) \geq n \cdot \Big(\frac{Ω(D(f))}{\log \mathsf{rk}(f)} -\log \mathsf{rk}(f)\Big )$, where here $D(f), D(f^{\oplus n})$ represent the deterministic communication complexity, and $\mathsf{rk}(f)$ is the rank of $f$. Our methods involve a new way to use information theory to reason about deterministic communication complexity.

An XOR Lemma for Deterministic Communication Complexity

TL;DR

The paper establishes a lower bound on the deterministic communication complexity of computing the parity in terms of the original complexity and the rank : . A central contribution is a novel information-theoretic method for deterministic protocols that avoids introducing errors, together with a subadditivity lemma: if has a monochromatic rectangle of size , then has a rectangle of size at least . This connection between rectangle structure at different levels, combined with a rank-based recursive decomposition, yields the main bound and advances understanding of how self-composition impacts deterministic communication complexity.

Abstract

We prove a lower bound on the communication complexity of computing the -fold xor of an arbitrary function , in terms of the communication complexity and rank of . We prove that , where here represent the deterministic communication complexity, and is the rank of . Our methods involve a new way to use information theory to reason about deterministic communication complexity.
Paper Structure (4 sections, 8 theorems, 21 equations)

This paper contains 4 sections, 8 theorems, 21 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notation
  3. Proof of \ref{['thm:subadditivity']}
  4. Proof of \ref{['main']}

Key Result

Theorem 1

$D(f^n) \geq \log C(f^{n}) \geq n \cdot (\sqrt{D(f)} - \log \log (|\mathcal{X}|\cdot |\mathcal{Y}|))$.

Theorems & Definitions (8)

  • Theorem 1: FKNN
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • Proposition 5
  • Lemma 6
  • Proposition 7
  • Lemma 8