An optimal lower bound on the communication complexity of Gap-Hamming-Distance

Amit Chakrabarti, Oded Regev

Research output: Chapter in Book/Report/Conference proceedingConference contribution


We prove an optimal Ω(n) lower bound on the randomized communication complexity of the much-studied Gap-Hamming-Distance problem. As a consequence, we obtain essentially optimal multi-pass space lower bounds in the data stream model for a number of fundamental problems, including the estimation of frequency moments. The Gap-Hamming-Distance problem is a communication problem, wherein Alice and Bob receive n-bit strings x and y, respectively. They are promised that the Hamming distance between x and y is either at least n/2+√n or at most n/2-√n, and their goal is to decide which of these is the case. Since the formal presentation of the problem by Indyk and Woodruff (FOCS, 2003), it had been conjectured that the naive protocol, which uses n bits of communication, is asymptotically optimal. The conjecture was shown to be true in several special cases, e.g., when the communication is deterministic, or when the number of rounds of communication is limited. The proof of our aforementioned result, which settles this conjecture fully, is based on a new geometric statement regarding correlations in Gaussian space, related to a result of C. Borell (1985). To prove this geometric statement, we show that random projections of not-too-small sets in Gaussian space are close to a mixture of translated normal variables.

Original languageEnglish (US)
Title of host publicationSTOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing
PublisherAssociation for Computing Machinery
Number of pages10
ISBN (Print)9781450306911
StatePublished - 2011
Event43rd ACM Symposium on Theory of Computing, STOC 2011 - San Jose, United States
Duration: Jun 6 2011Jun 8 2011

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017


Conference43rd ACM Symposium on Theory of Computing, STOC 2011
Country/TerritoryUnited States
CitySan Jose


  • communication complexity
  • corruption
  • data streams
  • gap-hamming-distance
  • gaussian noise correlation
  • lower bounds

ASJC Scopus subject areas

  • Software


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