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Sparse and Lopsided Set Disjointness via Information Theory

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Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX 2012, RANDOM 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7408))

Abstract

We study two natural variations of the set disjointness problem, arguably the most central problem in communication complexity.

For the k-sparse set disjointness problem, where the parties each hold a k-element subset of an n-element universe, we show a tight Θ(k logk) bound on the randomized one-way communication complexity. In addition, we present a slightly simpler proof of an O(k) upper bound on the general randomized communication complexity of this problem, due originally to Håstad and Wigderson.

For the lopsided set disjointness problem, we obtain a simpler proof of Pătraşcu’s breakthrough result, based on the information cost method of Bar-Yossef et al. The information-theoretic proof is both significantly simpler and intuitive; this is the first time the direct sum methodology based on information cost has been successfully adapted to the asymmetric communication setting. Our result shows that when Alice has a elements and Bob has b elements (a ≪ b) from an n-element universe, in any randomized protocol for disjointness, either Alice must communicate Ω(a) bits or Bob must communicate Ω(b) bits.

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Author information

Authors and Affiliations

  1. Yahoo! Research, 701 First Ave, Sunnyvale, CA, 94089, USA

    Anirban Dasgupta, Ravi Kumar & D. Sivakumar

Authors
  1. Anirban Dasgupta
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  2. Ravi Kumar
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  3. D. Sivakumar
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Editor information

Editors and Affiliations

  1. Department of Computer Science, Carnegie Mellon University, Gates Building 7203, 15213, Pittsburgh, PA, USA

    Anupam Gupta

  2. Department of Computer Science, University Kiel, Olshausenstraße 40, 24098, Kiel, Germany

    Klaus Jansen

  3. Centre Universitaire d’Informatique, University of Geneva, Battelle A, 7 route de Drize, 1227, Carouge, Switzerland

    José Rolim

  4. Department of Computer Science, Foundation School of Engineering and Applied Science, Columbia University, Amsterdam Avenue 1214, 10027-7003, New York, NY, USA

    Rocco Servedio

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© 2012 Springer-Verlag Berlin Heidelberg

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Dasgupta, A., Kumar, R., Sivakumar, D. (2012). Sparse and Lopsided Set Disjointness via Information Theory. In: Gupta, A., Jansen, K., Rolim, J., Servedio, R. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2012 2012. Lecture Notes in Computer Science, vol 7408. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32512-0_44

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  • DOI: https://doi.org/10.1007/978-3-642-32512-0_44

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-32511-3

  • Online ISBN: 978-3-642-32512-0

  • eBook Packages: Computer ScienceComputer Science (R0)

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