Gap Hamming Distance is a well-studied problem in communication complexity, in which Alice and Bob have to decide whether the Hamming distance between their respective n-bit inputs is less than \(n/2-\sqrt{n}\) or greater than \(n/2+\sqrt{n}\). We show that every k-round bounded-error communication protocol for this problem sends a message of at least Ω(n/(k 2logk)) bits. This lower bound has an exponentially better dependence on the number of rounds than the previous best bound, due to Brody and Chakrabarti. Our communication lower bound implies strong space lower bounds on algorithms for a number of data stream computations, such as approximating the number of distinct elements in a stream.


Communication Complexity Gap Hamming Distance Round Elimination Measure Concentration 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Joshua Brody
    • 1
  • Amit Chakrabarti
    • 1
  • Oded Regev
    • 2
  • Thomas Vidick
    • 3
  • Ronald de Wolf
    • 4
  1. 1.Department of Computer ScienceDartmouth CollegeHanover
  2. 2.Blavatnik School of Computer ScienceTel Aviv UniversityTel AvivIsrael
  3. 3.Department of Computer ScienceUCBerkeley
  4. 4.CWI Amsterdam 

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