Communication Lower Bounds Using Directional Derivatives

We study the set disjointness problem in the most powerful model of bounded-error communication, the k -party randomized number-on-the-forehead model. We show that set disjointness requires Ω(√n/2 k k ) bits of communication, where n is the size of the universe. Our lower bound generalizes to quantu...

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Bibliographic Details
Published inJournal of the ACM Vol. 61; no. 6; pp. 1 - 71
Main Author Sherstov, Alexander A.
Format Journal Article
LanguageEnglish
Published New York Association for Computing Machinery 01.11.2014
Subjects
Communication
Computer science
Derivatives
Error analysis
Lower bounds
Mathematical models
Mathematical problems
Optimization
Proof theory
Protocol
Studies
Universe
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ISSN0004-5411
1557-735X
DOI10.1145/2629334

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Summary:We study the set disjointness problem in the most powerful model of bounded-error communication, the k -party randomized number-on-the-forehead model. We show that set disjointness requires Ω(√n/2 k k ) bits of communication, where n is the size of the universe. Our lower bound generalizes to quantum communication, where it is essentially optimal. Proving this bound was a longstanding open problem even in restricted settings, such as one-way classical protocols with k =4 parties [Wigderson 1997]. The proof contributes a novel technique for lower bounds on multiparty communication, based on directional derivatives of protocols over the reals.
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ISSN:0004-5411
1557-735X
DOI:10.1145/2629334