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


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

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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)


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
  • Set membership
  • Bit-probe model
  • Fully-explicit data structures
  • Adaptive data structures
  • Error-correcting codes


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  1. Mirza Galib Anwarul Husain Baig and Deepanjan Kesh. Two new schemes in the bitprobe model. In M. Sohel Rahman, Wing-Kin Sung, and Ryuhei Uehara, editors, WALCOM: Algorithms and Computation - 12th International Conference, WALCOM 2018, Dhaka, Bangladesh, March 3-5, 2018, Proceedings, volume 10755 of Lecture Notes in Computer Science, pages 68-79. Springer, 2018. URL:
  2. Burton H. Bloom. Space/time trade-offs in hash coding with allowable errors. Communications of the ACM, 1970. Google Scholar
  3. R.C. Bose and D.K. Ray-Chaudhuri. On a class of error correcting binary group codes. Information and Control, 3(1):68-79, 1960. URL:
  4. Andrej Brodnik and J. Ian Munro. Membership in constant time and almost-minimum space. SIAM J. Comput., 28(5):1627–1640, May 1999. URL:
  5. H. Buhrman, P. B. Miltersen, J. Radhakrishnan, and S. Venkatesh. Are bitvectors optimal? SIAM J. Comput., 31(6):1723–1744, June 2002. URL:
  6. Michael L. Fredman, János Komlós, and Endre Szemerédi. Storing a sparse table with 0(1) worst case access time. J. ACM, 31(3):538–544, June 1984. URL:
  7. Mohit Garg and Jaikumar Radhakrishnan. Set membership with non-adaptive bit probes. CoRR, abs/1612.09388, 2016. URL:
  8. Mohit Garg and Jaikumar Radhakrishnan. Set membership with non-adaptive bit probes. In Proc. 34th Symposium on Theoretical Aspects of Computer Science, STACS 2017, March 8-11, 2017, Hannover, Germany, pages 38:1-38:13, 2017. URL:
  9. Mohit Garg and Jaikumar Radhakrishnan. Set membership with a few bit probes. In Proc. 26th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, pages 776-784, San Diego, CA, USA, January 4-6, 2015. URL:
  10. Venkatesan Guruswami. 15-859v: Introduction to coding theory, Spring 2010, CMU, 2010. URL:
  11. Alexis Hocquenghem. Codes correcteurs d'erreurs. Chiffres (Paris), 2:147-156, 1959. Google Scholar
  12. Deepanjan Kesh. Space Complexity of Two Adaptive Bitprobe Schemes Storing Three Elements. In Sumit Ganguly and Paritosh Pandya, editors, 38th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2018), volume 122 of Leibniz International Proceedings in Informatics (LIPIcs), pages 12:1-12:12, Dagstuhl, Germany, 2018. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL:
  13. Moshe Lewenstein, J. Ian Munro, Patrick K. Nicholson, and Venkatesh Raman. Improved explicit data structures in the bitprobe model. In Proc. 22nd Annual European Symposium on Algorithms, pages 630-641, Wroclaw, Poland, September 8-10, 2014. URL:
  14. A. Makhdoumi, S. Huang, M. Médard, and Y. Polyanskiy. On locally decodable source coding. In 2015 IEEE International Conference on Communications (ICC), pages 4394-4399, 2015. Google Scholar
  15. Marvin Minsky and Seymour A. Papert. Perceptrons: An Introduction to Computational Geometry. The MIT Press, 2017. Google Scholar
  16. Patrick K. Nicholson, Venkatesh Raman, and S. Srinivasa Rao. A Survey of Data Structures in the Bitprobe Model, pages 303-318. Springer Berlin Heidelberg, Berlin, Heidelberg, 2013. URL:
  17. Rasmus Pagh. Low redundancy in static dictionaries with constant query time. SIAM J. Comput., 31(2):353–363, February 2002. URL:
  18. Mihai Patrascu. Succincter. In Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science, FOCS ’08, page 305–313, USA, 2008. IEEE Computer Society. URL:
  19. Tilman Piesk. The 22 becs, 3-ary boolean functions. Wikiversity, 2016. URL:
  20. G. Pólya. Kombinatorische anzahlbestimmungen für gruppen, graphen und chemische verbindungen. Acta Math., 68:145-254, 1937. URL:
  21. J. Radhakrishnan, P. Sen, and S. Venkatesh. The quantum complexity of set membership. In Proceedings of the 41st Annual Symposium on Foundations of Computer Science, FOCS ’00, page 554, USA, 2000. IEEE Computer Society. Google Scholar
  22. Jaikumar Radhakrishnan, Venkatesh Raman, and S. Srinivasa Rao. Explicit deterministic constructions for membership in the bitprobe model. In Proc. 9th Annual European Symposium on Algorithms Algorithms, pages 290-299, Aarhus, Denmark, August 28-31, 2001. URL:
  23. Jaikumar Radhakrishnan, Smit Shah, and Saswata Shannigrahi. Data structures for storing small sets in the bitprobe model. In Proc. 18th Annual European Symposium on Algorithms Algorithms, pages 159-170, Liverpool, UK, September 6-8, 2010. URL:
  24. J. Howard Redfield. The theory of group-reduced distributions. American Journal of Mathematics, 49(3):433-455, 1927. URL:
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