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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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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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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.

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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

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.

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References

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