One-Sided Error Communication Complexity of Gap Hamming Distance

Authors Egor Klenin, Alexander Kozachinskiy

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Egor Klenin
  • Lomonosov Moscow State University, Moscow, Russia, Moscow, 1 Leninskiye Gory, Russia
Alexander Kozachinskiy
  • National Research University Higher School of Economics, Moscow, Russia, Moscow, 3 Kochnovsky Proezd, Russia

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Egor Klenin and Alexander Kozachinskiy. One-Sided Error Communication Complexity of Gap Hamming Distance. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 7:1-7:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


Assume that Alice has a binary string x and Bob a binary string y, both strings are of length n. Their goal is to output 0, if x and y are at least L-close in Hamming distance, and output 1, if x and y are at least U-far in Hamming distance, where L < U are some integer parameters known to both parties. If the Hamming distance between x and y lies in the interval (L, U), they are allowed to output anything. This problem is called the Gap Hamming Distance. In this paper we study public-coin one-sided error communication complexity of this problem. The error with probability at most 1/2 is allowed only for pairs at Hamming distance at least U. In this paper we determine this complexity up to factors logarithmic in L. The protocol we construct for the upper bound is simultaneous.

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
  • Communication Complexity
  • Gap Hamming Distance
  • one-sided error


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