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Abstract Voronoi-Like Graphs: Extending Delaunay’s Theorem and Applications

Author Evanthia Papadopoulou

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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Voronoi-like graph
  • abstract Voronoi diagram
  • Delaunay’s theorem
  • Voronoi tree
  • linear-time randomized algorithm
  • constraint Delaunay triangulation

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Abstract

Any system of bisectors (in the sense of abstract Voronoi diagrams) defines an arrangement of simple curves in the plane. We define Voronoi-like graphs on such an arrangement, which are graphs whose vertices are locally Voronoi. A vertex v is called locally Voronoi, if v and its incident edges appear in the Voronoi diagram of three sites. In a so-called admissible bisector system, where Voronoi regions are connected and cover the plane, we prove that any Voronoi-like graph is indeed an abstract Voronoi diagram. The result can be seen as an abstract dual version of Delaunay’s theorem on (locally) empty circles.
Further, we define Voronoi-like cycles in an admissible bisector system, and show that the Voronoi-like graph induced by such a cycle C is a unique tree (or a forest, if C is unbounded). In the special case where C is the boundary of an abstract Voronoi region, the induced Voronoi-like graph can be computed in expected linear time following the technique of [Junginger and Papadopoulou SOCG'18]. Otherwise, within the same time, the algorithm constructs the Voronoi-like graph of a cycle C′ on the same set (or subset) of sites, which may equal C or be enclosed by C. Overall, the technique computes abstract Voronoi (or Voronoi-like) trees and forests in linear expected time, given the order of their leaves along a Voronoi-like cycle. We show a direct application in updating a constraint Delaunay triangulation in linear expected time, after the insertion of a new segment constraint, simplifying upon the result of [Shewchuk and Brown CGTA 2015].

Cite As Get BibTex

Evanthia Papadopoulou. Abstract Voronoi-Like Graphs: Extending Delaunay’s Theorem and Applications. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 52:1-52:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.SoCG.2023.52

Author Details

Evanthia Papadopoulou
  • Faculty of Informatics, Università della Svizzera italiana, Lugano, Switzerland

Funding

Supported in part by the Swiss National Science Foundation, project 200021E-201356.

Acknowledgements

I would like to thank Kolja Junginger for preliminary discussions, numerous figures and comments on earlier versions of this work. I would also like to thank Franz Aurenhammer for valuable suggestions, and Elena Arseneva for constructive comments and figures.

References

  1. Alok Aggarwal, Leonidas J. Guibas, James B. Saxe, and Peter W. Shor. A linear-time algorithm for computing the Voronoi diagram of a convex polygon. Discrete and Computational Geometry, 4:591-604, 1989. URL: https://doi.org/10.1007/BF02187749.
  2. Franz Aurenhammer, Rolf Klein, and Der-Tsai Lee. Voronoi Diagrams and Delaunay Triangulations. World Scientific, 2013. URL: http://www.worldscientific.com/worldscibooks/10.1142/8685.
  3. Cecilia Bohler, Rolf Klein, and Chih-Hung Liu. Abstract Voronoi diagrams from closed bisecting curves. International Journal of Computational Geometry and Applications, 27(3):221-240, 2017. Google Scholar
  4. Cecilia Bohler, Rolf Klein, and Chih-Hung Liu. An efficient randomized algorithm for higher-order abstract Voronoi diagrams. Algorithmica, 81(6):2317-2345, 2019. Google Scholar
  5. L. Paul Chew. Building Voronoi diagrams for convex polygons in linear expected time. Technical report, Dartmouth College, Hanover, USA, 1990. Google Scholar
  6. B. Delaunay. Sur la sphère vide. A la memoire de Georges Voronoiï. Bulletin de l'Académie des Sciences de l'URSS. Classe des sciences mathématiques et naturelles, 6:793-800, 1934. Google Scholar
  7. H. Edelsbrunner and R. Seidel. Voronoi diagrams and Arrangements. Discrete and Comptational Geometry, 1:25-44, 1986. Google Scholar
  8. Kolja Junginger and Evanthia Papadopoulou. Deletion in Abstract Voronoi Diagrams in Expected Linear Time. In 34th International Symposium on Computational Geometry (SoCG), volume 99 of LIPIcs, pages 50:1-50:14, Dagstuhl, Germany, 2018. Google Scholar
  9. Kolja Junginger and Evanthia Papadopoulou. On tree-like abstract Voronoi diagrams in expected linear time. In CGWeek Young Researchers Forum (CG:YRF), 2019. Google Scholar
  10. Kolja Junginger and Evanthia Papadopoulou. Deletion in abstract Voronoi diagrams in expected linear time and related problems. arXiv:1803.05372v2 [cs.CG], 2020. To appear in Discrete and Computational Geometry. Google Scholar
  11. Rolf Klein. Concrete and Abstract Voronoi Diagrams, volume 400 of Lecture Notes in Computer Science. Springer-Verlag, 1989. Google Scholar
  12. Rolf Klein, Elmar Langetepe, and Z. Nilforoushan. Abstract Voronoi diagrams revisited. Computational Geometry: Theory and Applications, 42(9):885-902, 2009. Google Scholar
  13. Rolf Klein, Kurt Mehlhorn, and Stefan Meiser. Randomized incremental construction of abstract Voronoi diagrams. Computational Geometry: Theory and Applications, 3:157-184, 1993. Google Scholar
  14. Evanthia Papadopoulou. Abstract Voronoi-like Graphs: Extending Delaunay’s Theorem and Applications. arXiv:2303.06669 [cs.CG], March 2023. URL: https://doi.org/10.48550/arXiv.2303.06669.
  15. Micha Sharir and Pankaj K. Agarwal. Davenport-Schinzel sequences and their geometric applications. Cambridge university press, 1995. Google Scholar
  16. Jonathan Richard Shewchuk and Brielin C. Brown. Fast segment insertion and incremental construction of constrained Delaunay triangulations. Computational Geometry: Theory and Applications, 48(8):554-574, 2015. Google Scholar
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