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

TL;DR

This paper demonstrates a significant exponential separation between randomized and deterministic protocols in the Number-on-Forehead communication model by constructing a specific 3-player function with contrasting complexity requirements.

Contribution

It introduces an explicit 3-player function exhibiting an exponential gap in communication complexity between randomized and deterministic/no nondeterministic protocols.

Findings

Randomized protocols can compute the function with constant bits.

Deterministic and nondeterministic protocols require about (log N)^{1/3} bits.

The result extends recent progress on sets without 3-term arithmetic progressions.

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.