Hierarchical Categories in Colored Searching

Authors Peyman Afshani, Rasmus Killmann, Kasper Green Larsen

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Peyman Afshani
  • Aarhus University, Denmark
Rasmus Killmann
  • Aarhus University, Denmark
Kasper Green Larsen
  • Aarhus University, Denmark

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Peyman Afshani, Rasmus Killmann, and Kasper Green Larsen. Hierarchical Categories in Colored Searching. In 33rd International Symposium on Algorithms and Computation (ISAAC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 248, pp. 25:1-25:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


In colored range counting (CRC), the input is a set of points where each point is assigned a "color" (or a "category") and the goal is to store them in a data structure such that the number of distinct categories inside a given query range can be counted efficiently. CRC has strong motivations as it allows data structure to deal with categorical data. However, colors (i.e., the categories) in the CRC problem do not have any internal structure, whereas this is not the case for many datasets in practice where hierarchical categories exists or where a single input belongs to multiple categories. Motivated by these, we consider variants of the problem where such structures can be represented. We define two variants of the problem called hierarchical range counting (HCC) and sub-category colored range counting (SCRC) and consider hierarchical structures that can either be a DAG or a tree. We show that the two problems on some special trees are in fact equivalent to other well-known problems in the literature. Based on these, we also give efficient data structures when the underlying hierarchy can be represented as a tree. We show a conditional lower bound for the general case when the existing hierarchy can be any DAG, through reductions from the orthogonal vectors problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Categorical Data
  • Computational Geometry


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  1. Pankaj K. Agarwal. Range searching. In J. E. Goodman, J. O'Rourke, and C. Toth, editors, Handbook of Discrete and Computational Geometry. CRC Press, Inc., 2016. Google Scholar
  2. Stephen Alstrup, Gerth Stølting Brodal, and Theis Rauhe. New data structures for orthogonal range searching. In Proceedings of Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 198-207, 2000. Google Scholar
  3. Arturs Backurs and Piotr Indyk. Edit distance cannot be computed in strongly subquadratic time (unless seth is false). In Proceedings of the forty-seventh annual ACM symposium on Theory of computing, pages 51-58, 2015. Google Scholar
  4. Karl Bringmann. Why walking the dog takes time: Frechet distance has no strongly subquadratic algorithms unless seth fails. In 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pages 661-670. IEEE, 2014. Google Scholar
  5. Bernard Chazelle. Functional approach to data structures and its use in multidimensional searching. siamjour, 17(3):427-462, 1988. URL: https://doi.org/10.1137/0217026.
  6. Bernard Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM Journal of Computing, 17(3):427-462, 1988. Google Scholar
  7. Lech Duraj, Marvin Künnemann, and Adam Polak. Tight conditional lower bounds for longest common increasing subsequence. Algorithmica, 81(10):3968-3992, 2019. Google Scholar
  8. Johannes Fischer. Optimal succinctness for range minimum queries. In Proc. 9thLatin American Symposium on Theoretical Informatics (LATIN), pages 158-169, 2010. Google Scholar
  9. P. Gupta, R. Janardan, S. Rahul, and M. Smid. Computational geometry: generalized (or colored) intersection searching. In Handbook of Data Structures and Applications, chapter 67, pages 1-17. Chapman & Hall/CRC, 2017. Google Scholar
  10. P. Gupta, R. Janardan, and M. Smid. Further results on generalized intersection searching problems: Counting, reporting, and dynamization. Journal of Algorithms, 19(2):282-317, 1995. Google Scholar
  11. Dov Harel and Robert Endre Tarjan. Fast algorithms for finding nearest common ancestors. siam Journal on Computing, 13(2):338-355, 1984. Google Scholar
  12. Meng He and Serikzhan Kazi. Data Structures for Categorical Path Counting Queries. In Annual Symposium on Combinatorial Pattern Matching (CPM), volume 191, pages 15:1-15:17, 2021. Google Scholar
  13. Joseph JaJa, Christian W. Mortensen, and Qingmin Shi. Space-efficient and fast algorithms for multidimensional dominance reporting and counting. In Proc. 15thInternational Symposium on Algorithms and Computation (ISAAC), pages 558-568, 2004. Google Scholar
  14. R. Janardan and M. Lopez. Generalized intersection searching problems. International Journal of Computational Geometry & Applications, 3:39-69, 1993. Google Scholar
  15. Haim Kaplan, Natan Rubin, Micha Sharir, and Elad Verbin. Efficient colored orthogonal range counting. SIAM Journal of Computing, 38(3):982-1011, 2008. Google Scholar
  16. Kasper Green Larsen and Freek van Walderveen. Near-optimal range reporting structures for categorical data. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 265-277, 2013. Google Scholar
  17. Mihai Pǎtraşcu. Unifying the landscape of cell-probe lower bounds. In Proc. 49thProceedings of Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 434-443, 2008. Google Scholar
  18. Rahul Saladi. Approximate range counting revisited. Journal of Computational Geometry, 12(1), 2021. Google Scholar
  19. Daniel D. Sleator and Robert Endre Tarjan. A data structure for dynamic trees. Journal of Computer and System Sciences, 26(3):362-391, 1983. Google Scholar
  20. Ryan Williams. A new algorithm for optimal 2-constraint satisfaction and its implications. Theoretical Computer Science, 348(2):357-365, 2005. International Colloquium on Automata, Languages and Programming (ICALP). Google Scholar