Optimal Communication Rates for Zero-Error Distributed Simulation under Blackboard Communication Protocols

We study the distributed simulation problem where $n$ users aim to generate \emph{same} sequences of random coin flips. Some subsets of the users share an independent common coin which can be tossed multiple times, and there is a publicly seen blackboard through which the users communicate with each other. We show that when each coin is shared among subsets of size $k$, the communication rate (i.e., number of bits on blackboard per bit in generated sequence) is at least $\frac{n-k}{n-1}$. Moreover, if the size-$k$ subsets with common coins contain a path-connected cluster of topologically connected components, we propose a communication scheme which achieves the optimal rate $\frac{n-k}{n-1}$. Some key steps in analyzing the upper bounds rely on two different definitions of connectivity in hypergraphs, which may be of independent interest.