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The multiparty communication complexity of set disjointness

Published: 19 May 2012 Publication History

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.

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    cover image ACM Conferences
    STOC '12: Proceedings of the forty-fourth annual ACM symposium on Theory of computing
    May 2012
    1310 pages
    ISBN:9781450312455
    DOI:10.1145/2213977
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    Published: 19 May 2012

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    Author Tags

    1. circuit complexity
    2. direct product theorems
    3. multiparty communication complexity
    4. set disjointness problem

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    May 19 - 22, 2012
    New York, New York, USA

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