License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2020.46
URN: urn:nbn:de:0030-drops-126492
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12649/
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Golan, Shay ; Kociumaka, Tomasz ; Kopelowitz, Tsvi ; Porat, Ely ; Uznański, Przemysław

Improved Circular k-Mismatch Sketches

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Abstract

The shift distance sh(S₁,S₂) between two strings S₁ and S₂ of the same length is defined as the minimum Hamming distance between S₁ and any rotation (cyclic shift) of S₂. We study the problem of sketching the shift distance, which is the following communication complexity problem: Strings S₁ and S₂ of length n are given to two identical players (encoders), who independently compute sketches (summaries) sk(S₁) and sk(S₂), respectively, so that upon receiving the two sketches, a third player (decoder) is able to compute (or approximate) sh(S₁,S₂) with high probability.
This paper primarily focuses on the more general k-mismatch version of the problem, where the decoder is allowed to declare a failure if sh(S₁,S₂) > k, where k is a parameter known to all parties. Andoni et al. (STOC'13) introduced exact circular k-mismatch sketches of size Õ(k+D(n)), where D(n) is the number of divisors of n. Andoni et al. also showed that their sketch size is optimal in the class of linear homomorphic sketches.
We circumvent this lower bound by designing a (non-linear) exact circular k-mismatch sketch of size Õ(k); this size matches communication-complexity lower bounds. We also design (1± ε)-approximate circular k-mismatch sketch of size Õ(min(ε^{-2}√k, ε^{-1.5}√n)), which improves upon an Õ(ε^{-2}√n)-size sketch of Crouch and McGregor (APPROX'11).

BibTeX - Entry

@InProceedings{golan_et_al:LIPIcs:2020:12649,
  author =	{Shay Golan and Tomasz Kociumaka and Tsvi Kopelowitz and Ely Porat and Przemys{\l}aw Uznański},
  title =	{{Improved Circular k-Mismatch Sketches}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)},
  pages =	{46:1--46:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-164-1},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{176},
  editor =	{Jaros{\l}aw Byrka and Raghu Meka},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12649},
  URN =		{urn:nbn:de:0030-drops-126492},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2020.46},
  annote =	{Keywords: Hamming distance, k-mismatch, sketches, rotation, cyclic shift, communication complexity}
}

Keywords: Hamming distance, k-mismatch, sketches, rotation, cyclic shift, communication complexity
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)
Issue Date: 2020
Date of publication: 11.08.2020

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