Improved Separations between Nondeterministic and Randomized Multiparty Communication

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Abstract

We exhibit an explicit function f : {0, 1} n →{0,1} that can be computed by a nondeterministic number-on-forehead protocol communicating O(logn) bits, but that requires n Ω(1) bits of communication for randomized number-on-forehead protocols with k = δ·logn players, for any fixed δ< 1. Recent breakthrough results for the Set-Disjointness function (Sherstov, STOC ’08; Lee Shraibman, CCC ’08; Chattopadhyay Ada, ECCC ’08) imply such a separation but only when the number of players is k < loglogn.

We also show that for any k = A loglogn the above function f is computable by a small circuit whose depth is constant whenever A is a (possibly large) constant. Recent results again give such functions but only when the number of players is k < loglogn.