Space Efficient Two-Dimensional Orthogonal Colored Range Counting

Authors Younan Gao, Meng He

Thumbnail PDF


  • Filesize: 0.7 MB
  • 17 pages

Document Identifiers

Author Details

Younan Gao
  • Faculty of Computer Science, Dalhousie University, Halifax, Canada
Meng He
  • Faculty of Computer Science, Dalhousie University, Halifax, Canada

Cite AsGet BibTex

Younan Gao and Meng He. Space Efficient Two-Dimensional Orthogonal Colored Range Counting. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 46:1-46:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


In the two-dimensional orthogonal colored range counting problem, we preprocess a set, P, of n colored points on the plane, such that given an orthogonal query rectangle, the number of distinct colors of the points contained in this rectangle can be computed efficiently. For this problem, we design three new solutions, and the bounds of each can be expressed in some form of time-space tradeoff. By setting appropriate parameter values for these solutions, we can achieve new specific results with (the space costs are in words and ε is an arbitrary constant in (0,1)): - O(nlg³ n) space and O(√nlg^{5/2} n lg lg n) query time; - O(nlg² n) space and O(√nlg^{4+ε} n) query time; - O(n (lg² n)/(lg lg n)) space and O(√nlg^{5+ε} n) query time; - O(nlg n) space and O(n^{1/2+ε}) query time. A known conditional lower bound to this problem based on Boolean matrix multiplication gives some evidence on the difficulty of achieving near-linear space solutions with query time better than √n by more than a polylogarithmic factor using purely combinatorial approaches. Thus the time and space bounds in all these results are efficient. Previously, among solutions with similar query times, the most space-efficient solution uses O(nlg⁴ n) space to answer queries in O(√nlg⁸ n) time (SIAM. J. Comp. 2008). Thus the new results listed above all achieve improvements in space efficiency, while all but the last result achieve speed-up in query time as well.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Data structures design and analysis
  • 2D Colored orthogonal range counting
  • stabbing queries
  • geometric data structures


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Nikhil Bansal and Ryan Williams. Regularity lemmas and combinatorial algorithms. Theory Comput., 8(1):69-94, 2012. Google Scholar
  2. Timothy M. Chan. Speeding up the Four Russians Algorithm by About One More Logarithmic Factor. In SODA, pages 212-217, 2015. Google Scholar
  3. Timothy M. Chan, Qizheng He, and Yakov Nekrich. Further results on colored range searching. In Sergio Cabello and Danny Z. Chen, editors, 36th International Symposium on Computational Geometry, SoCG 2020, June 23-26, 2020, Zürich, Switzerland, volume 164 of LIPIcs, pages 28:1-28:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. Google Scholar
  4. Timothy M Chan, Kasper Green Larsen, and Mihai Pătraşcu. Orthogonal range searching on the ram, revisited. In 27th Symposium on Computational Geometry, pages 1-10. ACM, 2011. Google Scholar
  5. Timothy M Chan and Yakov Nekrich. Better data structures for colored orthogonal range reporting. In Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 627-636. SIAM, 2020. Google Scholar
  6. Moses Charikar, Surajit Chaudhuri, Rajeev Motwani, and Vivek R. Narasayya. Towards estimation error guarantees for distinct values. In PODS, pages 268-279, 2000. Google Scholar
  7. Bernard Chazelle. Filtering search: A new approach to query-answering. SIAM Journal on Computing, 15(3):703-724, 1986. Google Scholar
  8. Bernard Chazelle and Leonidas J Guibas. Fractional cascading: I. a data structuring technique. Algorithmica, 1(1-4):133-162, 1986. Google Scholar
  9. David R Clark and J Ian Munro. Efficient suffix trees on secondary storage. In Proceedings of the seventh annual ACM-SIAM symposium on Discrete algorithms, pages 383-391, 1996. Google Scholar
  10. Hicham El-Zein, J. Ian Munro, and Yakov Nekrich. Succinct color searching in one dimension. In Yoshio Okamoto and Takeshi Tokuyama, editors, 28th International Symposium on Algorithms and Computation, ISAAC 2017, December 9-12, 2017, Phuket, Thailand, volume 92 of LIPIcs, pages 30:1-30:11. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. Google Scholar
  11. Roberto Grossi and Søren Vind. Colored range searching in linear space. In Scandinavian Workshop on Algorithm Theory, pages 229-240. Springer, 2014. Google Scholar
  12. Prosenjit Gupta, Ravi Janardan, Saladi Rahul, and Michiel HM Smid. Computational geometry: Generalized (or colored) intersection searching. Handbook of Data Structures and Applications, pages 1042-1057, 2018. Google Scholar
  13. Prosenjit Gupta, Ravi Janardan, and Michiel Smid. Further results on generalized intersection searching problems: counting, reporting, and dynamization. Journal of Algorithms, 19(2):282-317, 1995. Google Scholar
  14. Prosenjit Gupta, Ravi Janardan, and Michiel Smid. Algorithms for generalized halfspace range searching and other intersection searching problems. Computational Geometry, 6(1):1-19, 1996. Google Scholar
  15. Yijie Han. Deterministic sorting in o (n log log n) time and linear space. In Proceedings of the thiry-fourth annual ACM symposium on Theory of computing, pages 602-608, 2002. Google Scholar
  16. Meng He and Serikzhan Kazi. Data structures for categorical path counting queries. In 32nd Annual Symposium on Combinatorial Pattern Matching, CPM 2021, To appear. Google Scholar
  17. Joseph JáJá, Christian W Mortensen, and Qingmin Shi. Space-efficient and fast algorithms for multidimensional dominance reporting and counting. In International Symposium on Algorithms and Computation, pages 558-568. Springer, 2004. Google Scholar
  18. Ravi Janardan and Mario Lopez. Generalized intersection searching problems. International Journal of Computational Geometry & Applications, 3(01):39-69, 1993. Google Scholar
  19. Haim Kaplan, Natan Rubin, Micha Sharir, and Elad Verbin. Efficient colored orthogonal range counting. SIAM Journal on Computing, 38(3):982-1011, 2008. Google Scholar
  20. Haim Kaplan, Micha Sharir, and Elad Verbin. Colored intersection searching via sparse rectangular matrix multiplication. In Proceedings of the twenty-second annual symposium on Computational geometry, pages 52-60, 2006. Google Scholar
  21. Ying Kit Lai, Chung Keung Poon, and Benyun Shi. Approximate colored range and point enclosure queries. Journal of Discrete Algorithms, 6(3):420-432, 2008. Google Scholar
  22. J. Ian Munro, Yakov Nekrich, and Sharma V. Thankachan. Range counting with distinct constraints. In Proceedings of the 27th Canadian Conference on Computational Geometry, CCCG 2015, Kingston, Ontario, Canada, August 10-12, 2015, pages 83-88. Queen’s University, Ontario, Canada, 2015. URL:
  23. Yakov Nekrich. Efficient range searching for categorical and plain data. ACM Transactions on Database Systems (TODS), 39(1):1-21, 2014. Google Scholar
  24. Saladi Rahul. Approximate range counting revisited. Journal of Computational Geometry, 12(1):40-69, 2021. Google Scholar
  25. Virginia Vassilevska Williams. Multiplying matrices faster than coppersmith-winograd. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 887-898, 2012. Google Scholar
  26. Huacheng Yu. An improved combinatorial algorithm for Boolean matrix multiplication. Inf. Comput., 261:240-247, 2018. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail