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.
@InProceedings{agrawal_et_al:LIPIcs.IPEC.2022.1, author = {Agrawal, Akanksha and Saurabh, Saket and Zehavi, Meirav}, title = {{A Finite Algorithm for the Realizabilty of a Delaunay Triangulation}}, booktitle = {17th International Symposium on Parameterized and Exact Computation (IPEC 2022)}, pages = {1:1--1:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-260-0}, ISSN = {1868-8969}, year = {2022}, volume = {249}, editor = {Dell, Holger and Nederlof, Jesper}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2022.1}, URN = {urn:nbn:de:0030-drops-173573}, doi = {10.4230/LIPIcs.IPEC.2022.1}, annote = {Keywords: Delaunay Triangulation, Delaunay Realization, Finite Algorithm, Integer Coordinate Realization} }
Feedback for Dagstuhl Publishing