Improved Separations between Nondeterministic and Randomized Multiparty Communication

We exhibit an explicit function f: {0, 1}ni¾?{0,1} that can be computed by a nondeterministic number-on-forehead protocol communicating O(logn) bits, but that requires ni¾?(1)bits of communication for randomized number-on-forehead protocols with k= i¾?·lognplayers, for any fixed i¾?< 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= Aloglognthe above function fis computable by a small circuit whose depth is constant whenever Ais a (possibly large) constant. Recent results again give such functions but only when the number of players is k< loglogn.

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