The multiparty communication complexity of set disjointness

  • Authors:

  • Alexander A. Sherstov

  • Affiliations:

  • University of California, Los Angeles, Los Angeles, CA, USA

  • Venue:
  • STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
  • Year:
  • 2012

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Abstract

We study the set disjointness problem in the number-on-the-forehead model of multiparty communication. (i) We prove that k-party set disjointness has communication complexity Omega(n/4k)1/4 in the randomized and nondeterministic models and Omega(n/4k)1/8 in the Merlin-Arthur model. These lower bounds are close to tight. Previous lower bounds (2007-2008) for k=3 parties were weaker than Omega(n/2k3)1/(k+1) in all three models. (ii) We prove that solving l instances of set disjointness requires l*Omega(n/4k)1/4 bits of communication, even to achieve correctness probability exponentially close to 1/2. This gives the first direct-product result for multiparty set disjointness, solving an open problem due to Beame, Pitassi, Segerlind, and Wigderson (2005). (iii) We construct a read-once {∧,∨}-circuit of depth 3 with exponentially small discrepancy for up to k≈(1/2)log n parties. This result is optimal with respect to depth and solves an open problem due to Beame and Huynh-Ngoc (FOCS '09), who gave a depth-6 construction. Applications to circuit complexity are given.