Optimal Communication Rates and Combinatorial Properties for Common Randomness Generation

Yanjun Han, Kedar Tatwawadi, Gowtham R. Kurri, Zhengqing Zhou, Vinod M. Prabhakaran, Tsachy Weissman

Abstract: We study the distributed simulation problem where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$n$</tex> players aim to generate same sequences of random coin flips, and some subsets of the players share an independent common coin which can be tossed multiple times. The players communicate with each other via a publicly seen blackboard. 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.