One-Sided Error Communication Complexity of Gap Hamming Distance
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.
Communication Complexity
Gap Hamming Distance
one-sided error
Theory of computation~Communication complexity
7:1-7:15
Regular Paper
Egor
Klenin
Egor Klenin
Lomonosov Moscow State University, Moscow, Russia, Moscow, 1 Leninskiye Gory, Russia
Alexander
Kozachinskiy
Alexander Kozachinskiy
National Research University Higher School of Economics, Moscow, Russia, Moscow, 3 Kochnovsky Proezd, Russia
https://orcid.org/0000-0002-9956-9023
Supported in part by RFBR grapnt 16-01-00362 and by the Russian Academic Excellence Project "5-100".
10.4230/LIPIcs.MFCS.2018.7
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Egor Klenin and Alexander Kozachinskiy
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