TY - GEN

T1 - A Finite Algorithm for the Realizabilty of a Delaunay Triangulation

AU - Agrawal, Akanksha

AU - Saurabh, Saket

AU - Zehavi, Meirav

N1 - Funding Information:
Funding Akanksha Agrawal: Supported by New Faculty Initiation Grant no. NFIG008972. Saket Saurabh: 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). Meirav Zehavi: Supported by Israel Science Foundation grant no. 1176/18, and United States – Israel Binational Science Foundation grant no. 2018302.
Publisher Copyright:
© Akanksha Agrawal, Saket Saurabh, and Meirav Zehavi.

PY - 2022/12/14

Y1 - 2022/12/14

N2 - 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. [19]. We design an nO(n)-time constructive algorithm for Delaunay Realization. In fact, our algorithm outputs sets of points with integer coordinates.

AB - 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. [19]. We design an nO(n)-time constructive algorithm for Delaunay Realization. In fact, our algorithm outputs sets of points with integer coordinates.

KW - Delaunay Realization

KW - Delaunay Triangulation

KW - Finite Algorithm

KW - Integer Coordinate Realization

UR - http://www.scopus.com/inward/record.url?scp=85144166941&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.IPEC.2022.1

DO - 10.4230/LIPIcs.IPEC.2022.1

M3 - Conference contribution

VL - 249

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 1:1-1:16

BT - 17th International Symposium on Parameterized and Exact Computation, IPEC 2022

A2 - Dell, Holger

A2 - Nederlof, Jesper

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 17th International Symposium on Parameterized and Exact Computation, IPEC 2022

Y2 - 7 September 2022 through 9 September 2022

ER -