Optimal Communication Rates and Combinatorial Properties for Common Randomness Generation

TL;DR

This paper characterizes the minimal communication needed for multiple players to generate identical random sequences using shared randomness and public communication, providing explicit algorithms and hypergraph-based insights.

Contribution

It offers a tight linear programming characterization of optimal communication rates and introduces explicit algorithms for distributed simulation in various hypergraph structures.

Findings

Optimal communication rates are characterized by linear programming.

Explicit algorithms are provided for hypergraphs with specific connectivity properties.

Achievability of rates extends from complete to sparser hypergraphs with certain topological features.

Abstract

We study common randomness generation problems where $n$ players aim to generate same sequences of random coin flips where some subsets of the players share an independent common coin which can be tossed multiple times, and there is a publicly seen blackboard through which the players communicate with each other. We provide a tight representation of the optimal communication rates via linear programming, and more importantly, propose explicit algorithms for the optimal distributed simulation for a wide class of hypergraphs. In particular, the optimal communication rate in complete hypergraphs is still achievable in sparser hypergraphs containing a path-connected cycle-free cluster of topologically connected components. Some key steps in analyzing the upper bounds rely on two different definitions of connectivity in hypergraphs, which may be of independent interest.