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.MFCS.2020.28
URN: urn:nbn:de:0030-drops-126965
URL: https://drops.dagstuhl.de/opus/volltexte/2020/12696/
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Dey, Palash ; Radhakrishnan, Jaikumar ; Velusamy, Santhoshini

Improved Explicit Data Structures in the Bit-Probe Model Using Error-Correcting Codes

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Abstract

We consider the bit-probe complexity of the set membership problem: represent an n-element subset S of an m-element universe as a succinct bit vector so that membership queries of the form "Is x ∈ S" can be answered using at most t probes into the bit vector. Let s(m,n,t) (resp. s_N(m,n,t)) denote the minimum number of bits of storage needed when the probes are adaptive (resp. non-adaptive). Lewenstein, Munro, Nicholson, and Raman (ESA 2014) obtain fully-explicit schemes that show that s(m,n,t) = 𝒪((2^t-1)m^{1/(t - min{2⌊log n⌋, n-3/2})}) for n ≥ 2,t ≥ ⌊log n⌋+1 . In this work, we improve this bound when the probes are allowed to be superlinear in n, i.e., when t ≥ Ω(nlog n), n ≥ 2, we design fully-explicit schemes that show that s(m,n,t) = 𝒪((2^t-1)m^{1/(t-{n-1}/{2^{t/(2(n-1))}})}), asymptotically (in the exponent of m) close to the non-explicit upper bound on s(m,n,t) derived by Radhakrishan, Shah, and Shannigrahi (ESA 2010), for constant n. In the non-adaptive setting, it was shown by Garg and Radhakrishnan (STACS 2017) that for a large constant n₀, for n ≥ n₀, s_N(m,n,3) ≥ √{mn}. We improve this result by showing that the same lower bound holds even for storing sets of size 2, i.e., s_N(m,2,3) ≥ Ω(√m).

BibTeX - Entry

@InProceedings{dey_et_al:LIPIcs:2020:12696,
  author =	{Palash Dey and Jaikumar Radhakrishnan and Santhoshini Velusamy},
  title =	{{Improved Explicit Data Structures in the Bit-Probe Model Using Error-Correcting Codes}},
  booktitle =	{45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)},
  pages =	{28:1--28:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-159-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{170},
  editor =	{Javier Esparza and Daniel Kr{\'a}ľ},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12696},
  URN =		{urn:nbn:de:0030-drops-126965},
  doi =		{10.4230/LIPIcs.MFCS.2020.28},
  annote =	{Keywords: Set membership, Bit-probe model, Fully-explicit data structures, Adaptive data structures, Error-correcting codes}
}

Keywords: Set membership, Bit-probe model, Fully-explicit data structures, Adaptive data structures, Error-correcting codes
Collection: 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020)
Issue Date: 2020
Date of publication: 18.08.2020


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