Improved Separations between Nondeterministic and Randomized Multiparty Communication

We exhibit an explicit function <i>f</i> : {0, 1}<sup>n</sup> →{0, 1} that can be computed by a nondeterministic number-on-forehead protocol communicating <i>O</i>(log<i>n</i>) bits, but that requires <i>n</i><sup>Ω(1)</sup> bits of communication for randomized number-on-forehead protocols with <i>k</i> = <i>Δ</i>·log<i>n</i> players, for any fixed <i>Δ</i> < 1. Recent breakthrough results for the Set-Disjointness function [Lee and Shraibman 2008; Chattopadhyay and Ada 2008] based on the work of Sherstov [2009; 2008a] imply such a separation but only when the number of players is <i>k</i> < loglog<i>n</i>. We also show that for any <i>k</i> = <i>A</i> ·loglog<i>n</i> the above function <i>f</i> is computable by a small circuit whose depth is constant whenever <i>A</i> is a (possibly large) constant. Recent results again give such functions but only when the number of players is <i>k</i> < loglog<i>n</i>.

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