Document Open Access Logo

A Finite Algorithm for the Realizabilty of a Delaunay Triangulation

Authors Akanksha Agrawal , Saket Saurabh , Meirav Zehavi

PDF

File

Thumbnail PDF
PDF
LIPIcs.IPEC.2022.1.pdf
  • Filesize: 1.1 MB
  • 16 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Delaunay Triangulation
  • Delaunay Realization
  • Finite Algorithm
  • Integer Coordinate Realization

Metrics

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

Abstract

The Delaunay graph of a point set P ⊆ ℝ² is the plane graph with the vertex-set P and the edge-set that contains {p,p'} if there exists a disc whose intersection with P is exactly {p,p'}. Accordingly, a triangulated graph G is Delaunay realizable if there exists a triangulation of the Delaunay graph of some P ⊆ ℝ², called a Delaunay triangulation of P, that is isomorphic to G. The objective of Delaunay Realization is to compute a point set P ⊆ ℝ² that realizes a given graph G (if such a P exists). Known algorithms do not solve Delaunay Realization as they are non-constructive. Obtaining a constructive algorithm for Delaunay Realization was mentioned as an open problem by Hiroshima et al. [Hiroshima et al., 2000]. We design an n^𝒪(n)-time constructive algorithm for Delaunay Realization. In fact, our algorithm outputs sets of points with integer coordinates.

Cite As Get BibTex

Akanksha Agrawal, Saket Saurabh, and Meirav Zehavi. A Finite Algorithm for the Realizabilty of a Delaunay Triangulation. In 17th International Symposium on Parameterized and Exact Computation (IPEC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 249, pp. 1:1-1:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.IPEC.2022.1

Author Details

Akanksha Agrawal
  • Indian Institute of Technology Madras, Chennai, India
Saket Saurabh
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
  • University of Bergen, Norway
Meirav Zehavi
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel

Funding

  • Agrawal, Akanksha: Supported by New Faculty Initiation Grant no. NFIG008972.
  • Saurabh, Saket: Supported by European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (no. 819416), and Swarnajayanti Fellowship (no. DST/SJF/MSA01/2017-18).
  • Zehavi, Meirav: Supported by Israel Science Foundation grant no. 1176/18, and United States – Israel Binational Science Foundation grant no. 2018302.

References

  1. Karim A. Adiprasito, Arnau Padrol, and Louis Theran. Universality theorems for inscribed polytopes and delaunay triangulations. Discrete & Computational Geometry, 54(2):412-431, 2015. Google Scholar
  2. Akanksha Agrawal, Saket Saurabh, and Meirav Zehavi. A finite algorithm for the realizabilty of a delaunay triangulation. arXiv, 2022. URL: https://doi.org/10.48550/ARXIV.2210.03932.
  3. Md. Ashraful Alam, Igor Rivin, and Ileana Streinu. Outerplanar graphs and Delaunay triangulations. In Proceedings of the 23rd Annual Canadian Conference on Computational (CCCG), 2011. Google Scholar
  4. Sanjeev Arora, Rong Ge, Ravindran Kannan, and Ankur Moitra. Computing a nonnegative matrix factorization - provably. In Proceedings of the 44th Annual ACM Symposium on Theory of Computing, STOC, pages 145-162, 2012. Google Scholar
  5. M. Artin. Algebra. Pearson Prentice Hall, 2011. Google Scholar
  6. Saugata Basu, Richard Pollack, and Marie-Françoise Roy. Algorithms in Real Algebraic Geometry (Algorithms and Computation in Mathematics). Springer-Verlag New York, Inc., Secaucus, NJ, USA, 2006. Google Scholar
  7. Mark de Berg, Otfried Cheong, Marc van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer-Verlag TELOS, 3rd ed. edition, 2008. Google Scholar
  8. Norishige Chiba, Kazunori Onoguchi, and Takao Nishizeki. Drawing plane graphs nicely. Acta Inf., 22(2):187-201, 1985. Google Scholar
  9. K. L. Clarkson and P. W. Shor. Applications of random sampling in computational geometry, II. Discrete Computational Geometry, 4:387-421, 1989. Google Scholar
  10. Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms (3. ed.). MIT Press, 2009. Google Scholar
  11. Hubert de Fraysseix, János Pach, and Richard Pollack. Small sets supporting fáry embeddings of planar graphs. In Proceedings of the 20th Annual ACM Symposium on Theory of Computing (STOC), pages 426-433, 1988. Google Scholar
  12. Giuseppe Di Battista and Luca Vismara. Angles of planar triangular graphs. SIAM Journal on Discrete Mathematics, 9(3):349-359, 1996. Google Scholar
  13. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  14. M. Dillencourt. Toughness and Delaunay triangulations. In Proceedings of the Third Annual Symposium on Computational Geometry, SoCG, pages 186-194, 1987. Google Scholar
  15. Michael. B. Dillencourt. Realizability of Delaunay triangulations. Information Processing Letters, 33:283-287, 1990. Google Scholar
  16. Michael B. Dillencourt and Warren D. Smith. Graph-theoretical conditions for inscribability and Delaunay realizability. Discrete Mathematics, 161(1-3):63-77, 1996. Google Scholar
  17. D. Yu. Grigor'ev and N. N. Vorobjov, Jr. Solving systems of polynomial inequalities in subexponential time. Journal of Symbolic Computation, 5:37-64, 1988. Google Scholar
  18. Leonidas J. Guibas, Donald E. Knuth, and Micha Sharir. Randomized incremental construction of Delaunay and Voronoi diagrams. Algorithmica, 7(1):381-413, 1992. Google Scholar
  19. Tetsuya Hiroshima, Yuichiro Miyamoto, and Kokichi Sugihara. Another proof of polynomial-time recognizability of Delaunay graphs. IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences, 83:627-638, 2000. Google Scholar
  20. Craig D Hodgson, Igor Rivin, and Warren D Smith. A characterization of convex hyperbolic polyhedra and of convex polyhedra inscribed in the sphere. Bulletin of the American Mathematical Society, 27:246-251, 1992. Google Scholar
  21. Jan Kratochvíl and Jivr'i Matouvsek. Intersection graphs of segments. J. Comb. Theory, Ser. B, 62(2):289-315, 1994. Google Scholar
  22. Timothy Lambert. An optimal algorithm for realizing a Delaunay triangulation. Information Processing Letters, 62(5):245-250, 1997. Google Scholar
  23. Colin McDiarmid and Tobias Müller. Integer realizations of disk and segment graphs. Journal of Combinatorial Theory, Series B, 103(1):114-143, 2013. Google Scholar
  24. Tobias Müller, Erik Jan van Leeuwen, and Jan van Leeuwen. Integer representations of convex polygon intersection graphs. SIAM J. Discrete Math., 27(1):205-231, 2013. Google Scholar
  25. Takao Nishizeki, Kazuyuki Miura, and Md. Saidur Rahman. Algorithms for drawing plane graphs. IEICE Transactions, 87-D(2):281-289, 2004. Google Scholar
  26. Yasuaki Oishi and Kokichi Sugihara. Topology-oriented divide-and-conquer algorithm for Voronoi diagrams. Graphical Models and Image Processing, 57:303-314, 1995. Google Scholar
  27. Atsuyuki Okabe, Barry Boots, and Kokichi Sugihara. Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. John Wiley & Sons, Inc., 1992. Google Scholar
  28. János Pach and Pankaj K. Agarwal. Combinatorial Geometry. Wiley-Interscience series in discrete mathematics and optimization. Wiley, New York, 1995. Google Scholar
  29. James Renegar. On the computational complexity and geometry of the first-order theory of the reals. Journal of Symbolic Computation, 13:255-352, 1992. Google Scholar
  30. Igor Rivin. Euclidean structures on simplicial surfaces and hyperbolic volume. Annals of Mathematics, 139:553-580, 1994. Google Scholar
  31. Kokichi Sugihara. Simpler proof of a realizability theorem on Delaunay triangulations. Information Processing Letters, 50:173-176, 1994. Google Scholar
  32. Kokichi Sugihara and Masao Iri. Construction of the Voronoi diagram for one million generators in single-precision arithmetic. Proceedings of the IEEE, 80:1471-1484, 1992. Google Scholar
  33. Kokichi Sugihara and Masao Iri. A robust topology-oriented incremental algorithm for Voronoi diagrams. International Journal of Computational Geometry & Applications, 4(02):179-228, 1994. Google Scholar
  34. William Thomas Tutte. How to draw a graph. Proceedings of the London Mathematical Society, 3(1):743-767, 1963. Google Scholar
  35. Hassler Whitney. Congruent Graphs and the Connectivity of Graphs, pages 61-79. Birkhäuser Boston, Boston, MA, 1992. Google Scholar
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