Coordination via Shared Randomness

We study a distributed sampling problem where a set of processors want to output correlated sequences of random variables with the help of a coordinator which has access to several independent sources of randomness. Each processor has access to a subset of these sources. When these subsets are pairwise disjoint (individually shared randomness model), we characterize the rate of communication required from the coordinator to the processors over a multicast link. We also give an upper bound on the communication rate for the randomness-on-the-forehead model where each processor observes all but one source of randomness. For the general model, we completely characterize the trade-off region between communication and shared randomness rates when all the processors wish to output the same random sequence.