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.


Random Mapping Communication Complexity Hellinger Distance Tight Lower Bound Parallel Repetition Theorem 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Anirban Dasgupta
    • 1
  • Ravi Kumar
    • 1
  • D. Sivakumar
    • 1
  1. 1.Yahoo! ResearchSunnyvaleUSA

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