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

Authors Palash Dey, Jaikumar Radhakrishnan, Santhoshini Velusamy



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Author Details

Palash Dey
  • Department of Computer Science and Engineering, Indian Institute of Technology Kharagpur, India
Jaikumar Radhakrishnan
  • School of Technology and Computer Science, Tata Institute of Fundamental Research, Mumbai, India
Santhoshini Velusamy
  • School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA

Acknowledgements

The authors would like to thank Madhu Sudan and the anonymous referees for several useful suggestions that helped improve the presentation of this paper.

Cite AsGet BibTex

Palash Dey, Jaikumar Radhakrishnan, and Santhoshini Velusamy. Improved Explicit Data Structures in the Bit-Probe Model Using Error-Correcting Codes. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 28:1-28:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.MFCS.2020.28

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).

Subject Classification

ACM Subject Classification
  • Theory of computation → Data compression
  • Theory of computation → Cell probe models and lower bounds
Keywords
  • Set membership
  • Bit-probe model
  • Fully-explicit data structures
  • Adaptive data structures
  • Error-correcting codes

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