Document Open Access Logo

Near-Optimal Dynamic Data Structures for Maximum Depth and Klee’s Measure of Boxes

Authors Sujoy Bhore , Subhash Suri , Jie Xue , Xiongxin Yang , Jiumu Zhu

PDF

File

Thumbnail PDF
PDF
LIPIcs.ICALP.2026.34.pdf
  • Filesize: 1.24 MB
  • 21 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • dynamic algorithms
  • maximum depth

Metrics

Abstract

We study two fundamental geometric problems on a dynamic set of n axis-parallel boxes in d-dimensional space. The maximum depth problem asks for the largest number of boxes that contain a common point, whereas Klee’s measure problem asks for the volume of the union of the boxes. We present fully dynamic exact data structures for both problems achieving Õ(n^{(d-1)/2}) amortized update time. This update time is optimal for an exact dynamic algorithm, up to logarithmic factors, assuming the Combinatorial k-Clique Hypothesis. Previously, matching bounds were established only for d = 1 [Imai and Asano, J. Algo.'83], and for d = 2 [Suri, Xue, Yang, and Zhu, SoCG'25].
Our approach integrates a classic grid-based partition framework with a novel charging analysis that controls the cost of structure-sensitive offline routines within each cell. This argument allows us to perform a global aggregation of the update time, by circumventing the worst-case costs associated with individual cell updates. We believe this technique may be of independent interest for other dynamic geometric problems.

Cite As Get BibTex

Sujoy Bhore, Subhash Suri, Jie Xue, Xiongxin Yang, and Jiumu Zhu. Near-Optimal Dynamic Data Structures for Maximum Depth and Klee’s Measure of Boxes. In 53rd International Colloquium on Automata, Languages, and Programming (ICALP 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 374, pp. 34:1-34:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ICALP.2026.34

Author Details

Sujoy Bhore
  • Indian Institute of Technology Bombay, Mumbai, India
Subhash Suri
  • University of California, Santa Barbara, CA, USA
Jie Xue
  • New York University Shanghai, China
Xiongxin Yang
  • University of California, Santa Barbara, CA, USA
Jiumu Zhu
  • New York University Shanghai, China

References

  1. Peyman Afshani and Timothy M. Chan. On approximate range counting and depth. Discrete Comput. Geom., 42(1):3-21, 2009. URL: https://doi.org/10.1007/S00454-009-9177-Z.
  2. Pankaj K. Agarwal, Hsien-Chih Chang, Subhash Suri, Allen Xiao, and Jie Xue. Dynamic geometric set cover and hitting set. ACM Trans. Algorithms, 18(4):40:1-40:37, 2022. URL: https://doi.org/10.1145/3551639.
  3. Helmut Alt and Ludmila Scharf. Computing the depth of an arrangement of axis-aligned rectangles in parallel. In Abstr. 26th Eur. Workshop Comput. Geom. (EuroCG), pages 33-36, 2010. Google Scholar
  4. Boris Aronov and Sariel Har-Peled. On approximating the depth and related problems. SIAM J. Comput., 38(3):899-921, 2008. URL: https://doi.org/10.1137/060669474.
  5. Jérémy Barbay, Timothy M. Chan, Gonzalo Navarro, and Pablo Pérez-Lantero. Maximum-weight planar boxes in time O(n²) (and better). Inf. Process. Lett., 114(8):437-445, 2014. Google Scholar
  6. Jon Louis Bentley. Algorithms for klee’s rectangle problem. Unpublished manuscript, 1977. Google Scholar
  7. Sujoy Bhore and Timothy M. Chan. Dynamic independent set of disks (and hypercubes) made easier. In Proc. 8th Sympos. Simplicity in Algorithms (SOSA), pages 485-495, 2025. URL: https://doi.org/10.1137/1.9781611978315.36.
  8. Sujoy Bhore and Timothy M. Chan. Fast static and dynamic approximation algorithms for geometric optimization problems: Piercing, independent set, vertex cover, and matching. In Proc. 36th ACM-SIAM Sympos. Discrete Algs. (SODA), pages 2357-2386, 2025. URL: https://doi.org/10.1137/1.9781611978322.79.
  9. Sujoy Bhore, Martin Nöllenburg, Csaba D. Tóth, and Jules Wulms. Fully dynamic maximum independent sets of disks in polylogarithmic update time. In Proc. 40th Int. Annu. Sympos. Comput. Geom. (SoCG), pages 19:1-19:16, 2024. URL: https://doi.org/10.4230/LIPIcs.SOCG.2024.19.
  10. Timothy M. Chan. Dynamic planar convex hull operations in near-logarithmic amortized time. J. ACM, 48(1):1-12, 2001. URL: https://doi.org/10.1145/363647.363652.
  11. Timothy M. Chan. Dynamic coresets. Discrete Comput. Geom., 42(3):469-488, 2009. URL: https://doi.org/10.1007/S00454-009-9165-3.
  12. Timothy M. Chan. A (slightly) faster algorithm for Klee’s measure problem. Comput. Geom., 43(3):243-250, 2010. URL: https://doi.org/10.1016/J.COMGEO.2009.01.007.
  13. Timothy M. Chan. Klee’s measure problem made easy. In Proc. 54th Annu. IEEE Sympos. Found. Comput. Sci. (FOCS), pages 410-419, 2013. URL: https://doi.org/10.1109/FOCS.2013.51.
  14. Timothy M. Chan. Dynamic geometric data structures via shallow cuttings. Discrete Comput. Geom., 64(4):1235-1252, 2020. URL: https://doi.org/10.1007/S00454-020-00229-5.
  15. Timothy M. Chan and Qizheng He. More dynamic data structures for geometric set cover with sublinear update time. J. Comput. Geom., 13(2):90-114, 2021. URL: https://doi.org/10.20382/JOCG.V13I2A6.
  16. Timothy M. Chan, Qizheng He, Subhash Suri, and Jie Xue. Dynamic geometric set cover, revisited. In Proc. 33rd ACM-SIAM Sympos. Discrete Algorithms (SODA), pages 3496-3528, 2022. URL: https://doi.org/10.1137/1.9781611977073.139.
  17. Timothy M. Chan and Konstantinos Tsakalidis. Dynamic orthogonal range searching on the RAM, revisited. J. Comput. Geom., 9(2):45-66, 2018. URL: https://doi.org/10.20382/JOCG.V9I2A5.
  18. Bernard Marie Chazelle and D. T. Lee. On a circle placement problem. Computing, 36(1-2):1-16, 1986. URL: https://doi.org/10.1007/BF02238188.
  19. Ludwig Danzer, Branko Grünbaum, and Victor Klee. Helly’s theorem and its relatives. In Convexity, Proc. Sympos. Pure Math., Vol. 7, Amer. Math. Soc., volume 7, page 101, 1963. Google Scholar
  20. David Dobkin and Subhash Suri. Maintenance of geometric extrema. J. ACM, 38(2):275-298, 1991. URL: https://doi.org/10.1145/103516.103518.
  21. D. Eppstein. Dynamic Euclidean minimum spanning trees and extrema of binary functions. Discrete Comput. Geom., 13(1):111-122, 1995. URL: https://doi.org/10.1007/BF02574030.
  22. Monika Henzinger, Stefan Neumann, and Andreas Wiese. Dynamic approximate maximum independent set of intervals, hypercubes and hyperrectangles. In Proc. 36th Int. Sympos. Comput. Geom. (SoCG), pages 51:1-51:14, 2020. URL: https://doi.org/10.4230/LIPIcs.SOCG.2020.51.
  23. John Hershberger and Subhash Suri. Off-line maintenance of planar configurations. J. Algorithms, 21(3):453-475, 1996. URL: https://doi.org/10.1006/JAGM.1996.0054.
  24. Hiroshi Imai and Takao Asano. Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane. J. Algorithms, 4(4):310-323, 1983. URL: https://doi.org/10.1016/0196-6774(83)90012-3.
  25. Victor Klee. Can the measure of ⋃_i = 1ⁿ [a_i, b_i] be computed in less than o(n log n) steps? Amer. Math. Monthly, 84(4):284-285, 1977. Google Scholar
  26. Jiri Matousek. Lectures on discrete geometry, volume 212. Springer Science & Business Media, 2013. Google Scholar
  27. Christian Worm Mortensen. Fully dynamic orthogonal range reporting on ram. SIAM J. Comput., 35(6):1494-1525, 2006. URL: https://doi.org/10.1137/S0097539703436722.
  28. Mark H. Overmars and Jan van Leeuwen. Maintenance of configurations in the plane. J. Comput. Syst. Sci., 23(2):166-204, 1981. URL: https://doi.org/10.1016/0022-0000(81)90012-X.
  29. Mark H. Overmars and Chee-Keng Yap. New upper bounds in Klee’s measure problem. SIAM J. Comput., 20(6):1034-1045, 1991. URL: https://doi.org/10.1137/0220065.
  30. Subhash Suri, Jie Xue, Xiongxin Yang, and Jiumu Zhu. Dynamic maximum depth of geometric objects. In Proc. 41st Int. Annu. Sympos. Comput. Geom. (SoCG), pages 77:1-77:13, 2025. URL: https://doi.org/10.4230/LIPIcs.SOCG.2025.77.
  31. Jan van Leeuwen and Derick Wood. The measure problem for rectangular ranges in d-space. J. Algorithms, 2(3):282-300, 1981. URL: https://doi.org/10.1016/0196-6774(81)90027-4.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact