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Explicit separations between randomized and deterministic Number-on-Forehead communication

Zander Kelley, Shachar Lovett, Raghu Meka

TL;DR

The paper addresses the separation between randomized and deterministic/non-deterministic Number-on-Forehead (NOF) communication, constructing an explicit 3-player function $D$ with constant randomized NOF complexity but $\Omega((\log N)^{1/3})$ deterministic/non-deterministic complexity. The approach introduces pseudorandomness against cubes and connects it to cylinder intersections, then develops a uniformity-from-spreadness machinery to analyze products of matrices and their correlations. A key technical engine is the theorem that the normalized product of spread, left-lower-bounded matrices is near-uniform, implemented via grid norms $U_{\ell,k}$. The construction of sparse pseudorandom sets from expander-colorings (notably using inner-product based expanders) yields an explicit, sparse $D$ that is easy for randomized NOF protocols yet hard for deterministic/non-deterministic NOF protocols, achieving a significant separation and linking NOF complexity to additive combinatorics issues such as 3-term progressions and corners. This work opens pathways to further explicit NOF separations (e.g., ExactlyN) and to potential multi-player generalizations through its expansion- and pseudorandomness-based framework.

Abstract

We study the power of randomness in the Number-on-Forehead (NOF) model in communication complexity. We construct an explicit 3-player function $f:[N]^3 \to \{0,1\}$, such that: (i) there exist a randomized NOF protocol computing it that sends a constant number of bits; but (ii) any deterministic or nondeterministic NOF protocol computing it requires sending about $(\log N)^{1/3}$ many bits. This exponentially improves upon the previously best-known such separation. At the core of our proof is an extension of a recent result of the first and third authors on sets of integers without 3-term arithmetic progressions into a non-arithmetic setting.

Explicit separations between randomized and deterministic Number-on-Forehead communication

TL;DR

The paper addresses the separation between randomized and deterministic/non-deterministic Number-on-Forehead (NOF) communication, constructing an explicit 3-player function with constant randomized NOF complexity but deterministic/non-deterministic complexity. The approach introduces pseudorandomness against cubes and connects it to cylinder intersections, then develops a uniformity-from-spreadness machinery to analyze products of matrices and their correlations. A key technical engine is the theorem that the normalized product of spread, left-lower-bounded matrices is near-uniform, implemented via grid norms . The construction of sparse pseudorandom sets from expander-colorings (notably using inner-product based expanders) yields an explicit, sparse that is easy for randomized NOF protocols yet hard for deterministic/non-deterministic NOF protocols, achieving a significant separation and linking NOF complexity to additive combinatorics issues such as 3-term progressions and corners. This work opens pathways to further explicit NOF separations (e.g., ExactlyN) and to potential multi-player generalizations through its expansion- and pseudorandomness-based framework.

Abstract

We study the power of randomness in the Number-on-Forehead (NOF) model in communication complexity. We construct an explicit 3-player function , such that: (i) there exist a randomized NOF protocol computing it that sends a constant number of bits; but (ii) any deterministic or nondeterministic NOF protocol computing it requires sending about many bits. This exponentially improves upon the previously best-known such separation. At the core of our proof is an extension of a recent result of the first and third authors on sets of integers without 3-term arithmetic progressions into a non-arithmetic setting.
Paper Structure (28 sections, 16 theorems, 140 equations)

This paper contains 28 sections, 16 theorems, 140 equations.

Table of Contents

  1. Introduction
  2. Proof overview
  3. Pseudorandomness against cubes
  4. From pseudorandomness against cubes to cylinder intersections
  5. Uniformity from spreadness
  6. Cylinder intersection closure
  7. Construction of sparse pseudorandom sets
  8. Paper organization.
  9. Acknowledgements.
  10. Preliminaries
  11. Notations.
  12. Norms and normalizations.
  13. Communication complexity.
  14. Uniformity from spreadness
  15. Grid norms
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $q$ be a prime power and $k$ a large enough constant ($k=34$ suffices). Let $N=q^k$, and identify $[N]$ with $\mathbb{F}_q^k$. Consider the following 3-player function: where $\left\langle\cdot ,\cdot \right\rangle$ denotes the standard inner-product in $\mathbb{F}_q^k$. Then:

Theorems & Definitions (64)

  • Theorem 1.1
  • Definition 2.1: Pseudorandom set with respect to large cubes
  • Theorem 2.2
  • Definition 2.3: Cylinder intersection
  • Theorem 2.4: Pseudorandom sets for cubes are pseudorandom for cylinder intersections
  • proof : Proof of \ref{['nondet-lb-psd-sparse']} from \ref{['psd-ci']}
  • Definition 2.5: Spreadness
  • Definition 2.6: Left lower-bounded
  • Definition 2.7: Uniformity
  • Theorem 2.8: Product of spread matrices is uniform
  • ...and 54 more