Multiparty Communication Complexity of AC

We prove n lower bounds on the multiparty communication complexity of AC functions in the number-on-forehead (NOF) model for up to Θ(logn) players. These are the first lower bounds for any AC function for ω(log logn) players. In particular we show that there are families of depth 3 read-once AC formulas having k-player randomized multiparty NOF communication complexity n/2. We show similar lower bounds for depth 4 read-once AC formulas that have nondeterministic communication complexity O(log n), yielding exponential separations between k-party nondeterministic and randomized communication complexity for AC functions. As a consequence of the latter bound, we obtain an n/2 lower bound on the k-party NOF communication complexity of set disjointness. This is non-trivial for up to Θ( √ logn) players which is significantly larger than the up to Θ(log logn) players allowed in the best previous lower bounds for multiparty set disjointness given by Lee and Shraibman [LS08] and Chattopadhyay and Ada [CA08] (though our complexity bounds themselves are not as strong as those in [LS08, CA08] for o(log logn) players). We derive these results by extending the k-party generalization in [CA08, LS08] of the pattern matrix method of Sherstov [She07, She08]. Using this technique, we derive a new sufficient criterion for strong communication complexity lower bounds based on functions having many diverse subfunctions that do not have good low-degree polynomial approximations. This criterion guarantees that such functions have orthogonalizing distributions that are “max-smooth” as opposed to the “min-smooth” orthogonalizing distributions used by Razborov and Sherstov [RS08] to analyze the sign-rank of AC. Research supported by NSF grant CCF-0514870 Research supported by a Vietnam Education Foundation Fellowship 1 Electronic Colloquium on Computational Complexity, Report No. 61 (2008)