Document Open Access Logo

Dynamic Colored Orthogonal Range Searching

Authors Timothy M. Chan , Zhengcheng Huang

PDF

File

Thumbnail PDF
PDF
LIPIcs.ESA.2021.28.pdf
  • Filesize: 0.66 MB
  • 13 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Data structures design and analysis
Keywords
  • Range searching
  • dynamic data structures
  • word RAM

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

In the colored orthogonal range reporting problem, we want a data structure for storing n colored points so that given a query axis-aligned rectangle, we can report the distinct colors among the points inside the rectangle. This natural problem has been studied in a series of papers, but most prior work focused on the static case. In this paper, we give a dynamic data structure in the 2D case which can answer queries in O(log^{1+o(1)} n + klog^{1/2+o(1)}n) time, where k denotes the output size (the number of distinct colors in the query range), and which can support insertions and deletions in O(log^{2+o(1)}n) time (amortized) in the standard RAM model. This is the first fully dynamic structure with polylogarithmic update time whose query cost per color reported is sublogarithmic (near √{log n}). We also give an alternative data structure with O(log^{1+o(1)} n + klog^{3/4+o(1)}n) query time and O(log^{3/2+o(1)}n) update time (amortized). We also mention extensions to higher constant dimensions.

Cite As Get BibTex

Timothy M. Chan and Zhengcheng Huang. Dynamic Colored Orthogonal Range Searching. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.ESA.2021.28

Author Details

Timothy M. Chan
  • Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA
Zhengcheng Huang
  • Department of Computer Science, University of Illinois at Urbana-Champaign, IL, USA

Funding

Supported by NSF Grant CCF-1814026.

Acknowledgements

We thank Saladi Rahul for helpful discussions.

References

  1. Pankaj K. Agarwal, Sathish Govindarajan, and S. Muthukrishnan. Range searching in categorical data: Colored range searching on grid. In Proc. 10th Annual European Symposium on Algorithms (ESA), pages 17-28, 2002. URL: https://doi.org/10.1007/3-540-45749-6_6.
  2. Stephen Alstrup, Thore Husfeldt, and Theis Rauhe. Marked ancestor problems. In Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 534-544, 1998. URL: https://doi.org/10.1109/SFCS.1998.743504.
  3. Lars Arge. The buffer tree: A technique for designing batched external data structures. Algorithmica, 37(1):1-24, 2003. URL: https://doi.org/10.1007/s00453-003-1021-x.
  4. Lars Arge and Jeffrey Scott Vitter. Optimal external memory interval management. SIAM J. Comput., 32(6):1488-1508, 2003. Google Scholar
  5. Panayiotis Bozanis, Nectarios Kitsios, Christos Makris, and Athanasios K. Tsakalidis. New upper bounds for generalized intersection searching problems. In Proc. 22nd International Colloquium on Automata, Languages and Programming (ICALP), pages 464-474, 1995. URL: https://doi.org/10.1007/3-540-60084-1_97.
  6. Timothy M. Chan, Qizheng He, and Yakov Nekrich. Further results on colored range searching. In Proc. 36th International Symposium on Computational Geometry (SoCG), pages 28:1-28:15, 2020. URL: https://doi.org/10.4230/LIPIcs.SoCG.2020.28.
  7. Timothy M. Chan and Yakov Nekrich. Better data structures for colored orthogonal range reporting. In Proc. 31st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 627-636, 2020. URL: https://doi.org/10.1137/1.9781611975994.38.
  8. Timothy M. Chan and Mihai Pătraşcu. Counting inversions, offline orthogonal range counting, and related problems. In Proc. 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 161-173, 2010. URL: https://doi.org/10.1137/1.9781611973075.15.
  9. Timothy M. Chan and Konstantinos Tsakalidis. Dynamic orthogonal range searching on the RAM, revisited. In Proc. 33rd International Symposium on Computational Geometry (SoCG), pages 28:1-28:13, 2017. URL: https://doi.org/10.4230/LIPIcs.SoCG.2017.28.
  10. Timothy M. Chan and Konstantinos Tsakalidis. Dynamic planar orthogonal point location in sublogarithmic time. In Proc. 34th International Symposium on Computational Geometry (SoCG), pages 25:1-25:15, 2018. URL: https://doi.org/10.4230/LIPIcs.SoCG.2018.25.
  11. Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag, 3rd edition, 2008. Google Scholar
  12. Hicham El-Zein, J. Ian Munro, and Yakov Nekrich. Succinct color searching in one dimension. In Proc. 28th International Symposium on Algorithms and Computation (ISAAC), pages 30:1-30:11, 2017. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2017.30.
  13. Travis Gagie, Juha Kärkkäinen, Gonzalo Navarro, and Simon J. Puglisi. Colored range queries and document retrieval. Theor. Comput. Sci., 483:36-50, 2013. URL: https://doi.org/10.1016/j.tcs.2012.08.004.
  14. Arnab Ganguly, J. Ian Munro, Yakov Nekrich, Rahul Shah, and Sharma V. Thankachan. Categorical range reporting with frequencies. In Proc. 22nd International Conference on Database Theory (ICDT), pages 9:1-9:19, 2019. URL: https://doi.org/10.4230/LIPIcs.ICDT.2019.9.
  15. Roberto Grossi and Søren Vind. Colored range searching in linear space. In Proc. 14th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT), pages 229-240, 2014. URL: https://doi.org/10.1007/978-3-319-08404-6_20.
  16. Prosenjit Gupta, Ravi Janardan, Saladi Rahul, and Michiel H. M. Smid. Computational geometry: Generalized (or colored) intersection searching. In Handbook of Data Structures and Applications, chapter 67, pages 1042-1057. CRC Press, 2nd edition, 2018. URL: https://www-users.cs.umn.edu/~sala0198/Papers/ds2-handbook.pdf.
  17. Prosenjit Gupta, Ravi Janardan, and Michiel H. M. Smid. Further results on generalized intersection searching problems: Counting, reporting, and dynamization. J. Algorithms, 19(2):282-317, 1995. URL: https://doi.org/10.1006/jagm.1995.1038.
  18. Prosenjit Gupta, Ravi Janardan, and Michiel H. M. Smid. Algorithms for generalized halfspace range searching and other intersection searching problems. Comput. Geom., 6:1-19, 1996. URL: https://doi.org/10.1016/0925-7721(95)00012-7.
  19. Prosenjit Gupta, Ravi Janardan, and Michiel H. M. Smid. A technique for adding range restrictions to generalized searching problems. Inf. Process. Lett., 64(5):263-269, 1997. URL: https://doi.org/10.1016/S0020-0190(97)00183-X.
  20. Prosenjit Gupta, Ravi Janardan, and Michiel H. M. Smid. Algorithms for some intersection searching problems involving circular objects. International Journal of Mathematical Algorithms, 1:35-52, 1999. Google Scholar
  21. Ravi Janardan and Mario A. Lopez. Generalized intersection searching problems. International Journal of Computational Geometry and Applications, 3(1):39-69, 1993. Google Scholar
  22. Haim Kaplan, Natan Rubin, Micha Sharir, and Elad Verbin. Efficient colored orthogonal range counting. SIAM J. Comput., 38(3):982-1011, 2008. URL: https://doi.org/10.1137/070684483.
  23. Haim Kaplan, Micha Sharir, and Elad Verbin. Colored intersection searching via sparse rectangular matrix multiplication. In Proc. 22nd ACM Symposium on Computational Geometry (SoCG), pages 52-60, 2006. URL: https://doi.org/10.1145/1137856.1137866.
  24. Kasper Green Larsen and Rasmus Pagh. I/O-efficient data structures for colored range and prefix reporting. In Proc. 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 583-592, 2012. URL: http://portal.acm.org/citation.cfm?id=2095165&CFID=63838676&CFTOKEN=79617016.
  25. Kasper Green Larsen and Freek van Walderveen. Near-optimal range reporting structures for categorical data. In Proc. 24th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 256-276, 2013. Google Scholar
  26. Kurt Mehlhorn and Stefan Näher. Dynamic fractional cascading. Algorithmica, 5(2):215-241, 1990. URL: https://doi.org/10.1007/BF01840386.
  27. Christian Worm Mortensen. Generalized static orthogonal range searching in less space. Technical report, IT University Technical Report Series 2003-33, 2003. Google Scholar
  28. 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.
  29. S. Muthukrishnan. Efficient algorithms for document retrieval problems. In Proc. 13th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 657-666, 2002. URL: https://doi.org/10.1145/545381.545469.
  30. Yakov Nekrich. Efficient range searching for categorical and plain data. ACM Trans. Database Syst., 39(1):9, 2014. URL: https://doi.org/10.1145/2543924.
  31. Yakov Nekrich and Jeffrey Scott Vitter. Optimal color range reporting in one dimension. In Proc. 21st Annual European Symposium Algorithms (ESA), pages 743-754, 2013. URL: https://doi.org/10.1007/978-3-642-40450-4_63.
  32. Manish Patil, Sharma V. Thankachan, Rahul Shah, Yakov Nekrich, and Jeffrey Scott Vitter. Categorical range maxima queries. In Proc. 33rd ACM Symposium on Principles of Database Systems (PODS), pages 266-277, 2014. URL: https://doi.org/10.1145/2594538.2594557.
  33. F. P. Preparata and M. I. Shamos. Computational Geometry: An Introduction. Springer-Verlag, 1985. Google Scholar
  34. Qingmin Shi and Joseph JáJá. Optimal and near-optimal algorithms for generalized intersection reporting on pointer machines. Inf. Process. Lett., 95(3):382-388, 2005. URL: https://doi.org/10.1016/j.ipl.2005.04.008.
  35. Peter van Emde Boas. Preserving order in a forest in less than logarithmic time and linear space. Inf. Process. Lett., 6(3):80-82, 1977. URL: https://doi.org/10.1016/0020-0190(77)90031-X.
  36. Bryan T. Wilkinson. Amortized bounds for dynamic orthogonal range reporting. In Proc. 22th Annual European Symposium on Algorithms (ESA), pages 842-856, 2014. URL: https://doi.org/10.1007/978-3-662-44777-2_69.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact