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A Simple Dynamization of Trapezoidal Point Location in Planar Subdivisions

Authors Milutin Brankovic, Nikola Grujic, André van Renssen , Martin P. Seybold

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Author Details

Milutin Brankovic
  • University of Sydney, Australia
Nikola Grujic
  • University of Sydney, Australia
André van Renssen
  • University of Sydney, Australia
Martin P. Seybold
  • University of Sydney, Australia

Funding

  • Seybold, Martin P.: Supported under the Australian Research Council Discovery Projects funding scheme (project number DP180102870).

Acknowledgements

We want to thank the University of Sydney’s undergraduate Research Internship Program, which fostered the pursuit of this topic.

Cite As Get BibTex

Milutin Brankovic, Nikola Grujic, André van Renssen, and Martin P. Seybold. A Simple Dynamization of Trapezoidal Point Location in Planar Subdivisions. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 18:1-18:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.18

Abstract

We study how to dynamize the Trapezoidal Search Tree (TST) - a well known randomized point location structure for planar subdivisions of kinetic line segments.
Our approach naturally extends incremental leaf-level insertions to recursive methods and allows adaptation for the online setting. The dynamization carries over to the Trapezoidal Search DAG (TSD), which has linear size and logarithmic point location costs with high probability. On a set S of non-crossing segments, each TST update performs expected 𝒪(log²|S|) operations and each TSD update performs expected 𝒪(log |S|) operations.
We demonstrate the practicality of our method with an open-source implementation, based on the Computational Geometry Algorithms Library, and experiments on the update performance.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
Keywords
  • Dynamization
  • Trapezoidal Search Tree
  • Trapezoidal Search DAG
  • Backward Analysis
  • Point Location
  • Planar Subdivision
  • Treap
  • Order-maintenance

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References

  1. Pankaj K. Agarwal, Julien Basch, Mark de Berg, Leonidas J. Guibas, and John Hershberger. Lower bounds for kinetic planar subdivisions. Discrete & Computational Geometry, 24(4):721-733, 2000. URL: https://doi.org/10.1007/s004540010060.
  2. Pankaj K. Agarwal, Jeff Erickson, and Leonidas J. Guibas. Kinetic binary space partitions for intersecting segments and disjoint triangles. In Proceedings of the 9th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 107-116, 1998. URL: http://dl.acm.org/citation.cfm?id=314613.
  3. Lars Arge, Gerth Stølting Brodal, and Loukas Georgiadis. Improved dynamic planar point location. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 305-314, 2006. URL: https://doi.org/10.1109/FOCS.2006.40.
  4. Daniel Bahrdt and Martin P. Seybold. Rational points on the unit sphere: Approximation complexity and practical constructions. In Proceedings of the 2017 ACM on International Symposium on Symbolic and Algebraic Computation (ISSAC), pages 29-36, 2017. URL: https://doi.org/10.1145/3087604.3087639.
  5. Hanna Baumgarten, Hermann Jung, and Kurt Mehlhorn. Dynamic point location in general subdivisions. Journal of Algorithms, 17(3):342-380, 1994. URL: https://doi.org/10.1006/jagm.1994.1040.
  6. Michael A. Bender, Jeremy T. Fineman, Seth Gilbert, Tsvi Kopelowitz, and Pablo Montes. File maintenance: When in doubt, change the layout! In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1503-1522, 2017. URL: https://doi.org/10.1137/1.9781611974782.98.
  7. Timothy M. Chan and Yakov Nekrich. Towards an optimal method for dynamic planar point location. In Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 390-409, 2015. URL: https://doi.org/10.1109/FOCS.2015.31.
  8. Siu-Wing Cheng and Ravi Janardan. New results on dynamic planar point location. SIAM Journal on Computing, 21(5):972-999, 1992. URL: https://doi.org/10.1137/0221057.
  9. Y. Chiang and R. Tamassia. Dynamic algorithms in computational geometry. Proceedings of the IEEE, 80(9):1412-1434, 1992. URL: https://doi.org/10.1109/5.163409.
  10. Yi-Jen Chiang and Roberto Tamassia. Dynamization of the trapezoid method for planar point location in monotone subdivisions. International Journal of Computational Geometry & Applications, 2(3):311-333, 1992. URL: https://doi.org/10.1142/S0218195992000184.
  11. Mark de Berg, Otfried Cheong, Marc J. van Kreveld, and Mark H. Overmars. Computational Geometry: Algorithms and Applications, 3rd Edition. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-77974-2.
  12. Michael T. Goodrich and Roberto Tamassia. Dynamic trees and dynamic point location. SIAM Journal on Computing, 28(2):612-636, 1998. URL: https://doi.org/10.1137/S0097539793254376.
  13. Michael Hemmer, Michal Kleinbort, and Dan Halperin. Optimal randomized incremental construction for guaranteed logarithmic planar point location. Computational Geometry: Theory and Applications, 58:110-123, 2016. URL: https://doi.org/10.1016/j.comgeo.2016.07.006.
  14. Alex Hobé, Daniel Vogler, Martin P. Seybold, Anozie Ebigbo, Randolph R. Settgast, and Martin O. Saar. Estimating fluid flow rates through fracture networks using combinatorial optimization. Advances in Water Resources, 122:85-97, 2018. URL: https://doi.org/10.1016/j.advwatres.2018.10.002.
  15. Ketan Mulmuley. A fast planar partition algorithm, I. Journal of Symbolic Computation, 10(3-4):253-280, 1990. URL: https://doi.org/10.1016/S0747-7171(08)80064-8.
  16. Ketan Mulmuley. Computational Geometry: An Introduction Through Randomized Algorithms. Prentice Hall, 1994. Google Scholar
  17. J. Ian Munro and Yakov Nekrich. Dynamic planar point location in external memory. In Proceedings of the 35th International Symposium on Computational Geometry (SoCG), pages 52:1-52:15, 2019. URL: https://doi.org/10.4230/LIPIcs.SoCG.2019.52.
  18. Eunjin Oh and Hee-Kap Ahn. Point location in dynamic planar subdivisions. In Proceedings of the 34th International Symposium on Computational Geometry (SoCG), pages 63:1-63:14, 2018. URL: https://doi.org/10.4230/LIPIcs.SoCG.2018.63.
  19. Otfried Schwarzkopf. Dynamic maintenance of geometric structures made easy. In Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 197-206, 1991. URL: https://doi.org/10.1109/SFCS.1991.185369.
  20. Raimund Seidel. A simple and fast incremental randomized algorithm for computing trapezoidal decompositions and for triangulating polygons. Computational Geometry: Theory and Applications, 1:51-64, 1991. URL: https://doi.org/10.1016/0925-7721(91)90012-4.
  21. Raimund Seidel and Cecilia R. Aragon. Randomized search trees. Algorithmica, 16(4-5):464-497, 1996. URL: https://doi.org/10.1007/BF01940876.
  22. Martin P. Seybold. Robust map matching for heterogeneous data via dominance decompositions. In Proceedings of the 2017 SIAM International Conference on Data Mining (SDM), pages 813-821, 2017. URL: https://doi.org/10.1137/1.9781611974973.91.
  23. The CGAL Project. CGAL User and Reference Manual. CGAL Editorial Board, 4.14.1 edition, 2019. URL: https://doc.cgal.org/4.14.1/Manual/packages.html.
  24. Athanasios K. Tsakalidis. Maintaining order in a generalized linked list. Acta Informatica, 21:101-112, 1984. URL: https://doi.org/10.1007/BF00289142.
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