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A Finite Algorithm for the Realizabilty of a Delaunay Triangulation

Authors Akanksha Agrawal , Saket Saurabh , Meirav Zehavi

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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.

Cite AsGet 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

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.

Subject Classification

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

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