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A Simple Integer Successor-Delete Data Structure

Author Gerth Stølting Brodal

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LIPIcs.SEA.2025.8.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Data structures design and analysis
Keywords
  • Successor queries
  • deletions
  • interval union-find
  • union-find

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Abstract

We consider a simple decremental data structure for maintaining a set of integers, that supports initializing the set to {1,2,…,n} followed by d deletions and s successor queries in arbitrary order in total 𝒪(n+d+s⋅(1+log_{max(2,s/n)} min(s,n))) time. The data structure consists of a single array of n integers. A straightforward modification allows the data structure to also support p predecessor and r range queries, with a total output k, in total 𝒪(n+d+k+q ⋅ (1+log_{max(2,q/n)} min(q,n))) time, where q = s+p+r. The data structure is essentially a special case of the classic union-find data structure with path compression but with unweighted linking (i.e., without linking by rank or size), that is known to achieve logarithmic amortized time bounds (Tarjan and van Leeuwen, 1984). In this paper we study the efficiency of this simple data structure, and compare it to other, theoretically superior, data structures.

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Gerth Stølting Brodal. A Simple Integer Successor-Delete Data Structure. In 23rd International Symposium on Experimental Algorithms (SEA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 338, pp. 8:1-8:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SEA.2025.8

Author Details

Gerth Stølting Brodal
  • Department of Computer Science, Aarhus University, Denmark

Funding

Supported by Independent Research Fund Denmark, grant 9131-00113B.

Supplementary Materials

References

  1. Georgy M. Adelson-Velsky and Evgenii M. Landis. An algorithm for the organization of information. Proceedings of the USSR Academy of Sciences (in Russian), 146:263-266, 1962. Google Scholar
  2. Alfred V. Aho, John E. Hopcroft, and Jeffrey D. Ullman. The Design and Analysis of Computer Algorithms. Addison-Wesley, 1974. Google Scholar
  3. Paul Beame and Faith E. Fich. Optimal bounds for the predecessor problem and related problems. Journal of Computer and System Sciences, 65(1):38-72, 2002. URL: https://doi.org/10.1006/JCSS.2002.1822.
  4. Edsger W. Dijkstra. A Discipline of Programming. Prentice-Hall, 1976. Google Scholar
  5. Patrick Dinklage, Johannes Fischer, and Alexander Herlez. Engineering predecessor data structures for dynamic integer sets. In David Coudert and Emanuele Natale, editors, 19th International Symposium on Experimental Algorithms, SEA 2021, June 7-9, 2021, Nice, France, volume 190 of LIPIcs, pages 7:1-7:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.SEA.2021.7.
  6. Michael J. Fischer. Efficiency of equivalence algorithms. In Raymond E. Miller and James W. Thatcher, editors, Proceedings of a symposium on the Complexity of Computer Computations, held March 20-22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, USA, The IBM Research Symposia Series, pages 153-167. Plenum Press, New York, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_14.
  7. Gianni Franceschini and Roberto Grossi. Optimal worst-case operations for implicit cache-oblivious search trees. In Frank K. H. A. Dehne, Jörg-Rüdiger Sack, and Michiel H. M. Smid, editors, Algorithms and Data Structures, 8th International Workshop, WADS 2003, Ottawa, Ontario, Canada, July 30 - August 1, 2003, Proceedings, volume 2748 of Lecture Notes in Computer Science, pages 114-126. Springer, 2003. URL: https://doi.org/10.1007/978-3-540-45078-8_11.
  8. Gianni Franceschini and Roberto Grossi. Optimal implicit dictionaries over unbounded universes. Theory of Computing Systems, 39(2):321-345, 2006. URL: https://doi.org/10.1007/S00224-005-1167-9.
  9. Harold N. Gabow and Robert Endre Tarjan. A linear-time algorithm for a special case of disjoint set union. Journal of Computer and System Sciences, 30(2):209-221, 1985. URL: https://doi.org/10.1016/0022-0000(85)90014-5.
  10. Bernard A. Galler and Michael J. Fischer. An improved equivalence algorithm. Communications of the ACM, 7(5):301-303, 1964. URL: https://doi.org/10.1145/364099.364331.
  11. Ashish Goel, Sanjeev Khanna, Daniel H. Larkin, and Robert Endre Tarjan. Disjoint set union with randomized linking. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1005-1017. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.75.
  12. Leonidas J. Guibas and Robert Sedgewick. A dichromatic framework for balanced trees. In 19th Annual Symposium on Foundations of Computer Science, Ann Arbor, Michigan, USA, 16-18 October 1978, pages 8-21. IEEE Computer Society, 1978. URL: https://doi.org/10.1109/SFCS.1978.3.
  13. Giuseppe F. Italiano and Rajeev Raman. Topics in data structures. In Mikhail J. Atallah, editor, Algorithms and Theory of Computation Handbook, Chapman & Hall/CRC Applied Algorithms and Data Structures series. CRC Press, 1999. URL: https://doi.org/10.1201/9781420049503-C6.
  14. Tianxiao Li, Jingxun Liang, Huacheng Yu, and Renfei Zhou. Dynamic "Succincter". In 64th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2023, Santa Cruz, CA, USA, November 6-9, 2023, pages 1715-1733. IEEE, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00104.
  15. J. Ian Munro. An implicit data structure supporting insertion, deletion, and search in O(log² n) time. Journal of Computer and System Sciences, 33(1):66-74, 1986. URL: https://doi.org/10.1016/0022-0000(86)90043-7.
  16. Mihai Pătraşcu and Mikkel Thorup. Time-space trade-offs for predecessor search. In Jon M. Kleinberg, editor, Proceedings of the 38th Annual ACM Symposium on Theory of Computing, Seattle, WA, USA, May 21-23, 2006, pages 232-240. ACM, 2006. URL: https://doi.org/10.1145/1132516.1132551.
  17. Mihai Pătraşcu and Mikkel Thorup. Dynamic integer sets with optimal rank, select, and predecessor search. In 55th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2014, Philadelphia, PA, USA, October 18-21, 2014, pages 166-175. IEEE Computer Society, 2014. URL: https://doi.org/10.1109/FOCS.2014.26.
  18. Md. Mostofa Ali Patwary, Jean R. S. Blair, and Fredrik Manne. Experiments on union-find algorithms for the disjoint-set data structure. In Paola Festa, editor, Experimental Algorithms, 9th International Symposium, SEA 2010, Ischia Island, Naples, Italy, May 20-22, 2010. Proceedings, volume 6049 of Lecture Notes in Computer Science, pages 411-423. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-13193-6_35.
  19. Giulio Ermanno Pibiri and Rossano Venturini. Dynamic elias-fano representation. In Juha Kärkkäinen, Jakub Radoszewski, and Wojciech Rytter, editors, 28th Annual Symposium on Combinatorial Pattern Matching, CPM 2017, July 4-6, 2017, Warsaw, Poland, volume 78 of LIPIcs, pages 30:1-30:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL: https://doi.org/10.4230/LIPICS.CPM.2017.30.
  20. Daniel Dominic Sleator and Robert Endre Tarjan. Self-adjusting binary search trees. Journal of the ACM, 32(3):652-686, 1985. URL: https://doi.org/10.1145/3828.3835.
  21. Robert Endre Tarjan. Efficiency of a good but not linear set union algorithm. Journal of the ACM, 22(2):215-225, 1975. URL: https://doi.org/10.1145/321879.321884.
  22. Robert Endre Tarjan. Amortized computational complexity. SIAM Journal on Algebraic Discrete Methods, 6(2):306-318, 1985. URL: https://doi.org/10.1137/0606031.
  23. Robert Endre Tarjan and Jan van Leeuwen. Worst-case analysis of set union algorithms. Journal of the ACM, 31(2):245-281, 1984. URL: https://doi.org/10.1145/62.2160.
  24. Theodorus Petrus van der Weide. Datastructures: An Axiomatic Approach and the Use of Binomial Trees in Developing and Analyzing Algorithms. PhD thesis, Mathematisch Centrum, Amsterdam, 1980. URL: https://www.cs.ru.nl/Th.P.vanderWeide/docs/1980-Weide-AxAppr.pdf.
  25. Peter van Emde Boas. Preserving order in a forest in less than logarithmic time and linear space. Information Processing Letters, 6(3):80-82, 1977. URL: https://doi.org/10.1016/0020-0190(77)90031-X.
  26. Peter van Emde Boas, R. Kaas, and E. Zijlstra. Design and implementation of an efficient priority queue. Mathematical Systems Theory, 10:99-127, 1977. URL: https://doi.org/10.1007/BF01683268.
  27. Jan van Leeuwen and Theo van der Weide. Alternative path compression techniques. Technical Report RUU-CS-77-3, Department of Computer Science, University of Utrecht, The Neterlands, 1977. URL: https://dspace.library.uu.nl/handle/1874/15885.
  28. J. W. J. Williams. Algorithm 232: Heapsort. Communications of the ACM, 7(6):347-348, 1964. Google Scholar
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