The farthest color voronoi diagram in the plane

The farthest-color Voronoi diagram (FCVD) is defined on a set of n points in the plane, where each point is labeled with one of m colors. The colored points constitute a family P of m clusters (sets) of points in the plane whose farthest-site Voronoi diagram is the FCVD. The diagram finds applicatio...

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Autores: Mantas, Ioannis, Papadopoulou, Evanthia|||0000-0003-0144-7384, Silveira, Rodrigo Ignacio|||0000-0003-0202-4543, Wang, Zeyu|||0009-0004-4207-198X
Tipo de recurso: artículo
Fecha de publicación:2025
País:España
Institución:Universitat Politècnica de Catalunya (UPC)
Repositorio:UPCommons. Portal del coneixement obert de la UPC
Idioma:inglés
OAI Identifier:oai:upcommons.upc.edu:2117/433637
Acceso en línea:https://hdl.handle.net/2117/433637
https://dx.doi.org/10.1007/s00453-025-01311-1
Access Level:acceso abierto
Palabra clave:Farthest-site Voronoi diagram
Color Voronoi diagram
Point clusters
Color spanning disk
Straddles
Divide and conquer
Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi matemàtica
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spelling The farthest color voronoi diagram in the planeMantas, IoannisPapadopoulou, Evanthia|||0000-0003-0144-7384Silveira, Rodrigo Ignacio|||0000-0003-0202-4543Wang, Zeyu|||0009-0004-4207-198XFarthest-site Voronoi diagramColor Voronoi diagramPoint clustersColor spanning diskStraddlesDivide and conquerÀrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi matemàticaThe farthest-color Voronoi diagram (FCVD) is defined on a set of n points in the plane, where each point is labeled with one of m colors. The colored points constitute a family P of m clusters (sets) of points in the plane whose farthest-site Voronoi diagram is the FCVD. The diagram finds applications in problems related to facility location, shape matching, data imprecision, and others. In this paper we present structural properties of the FCVD, refine its combinatorial complexity bounds, and present efficient algorithms for its construction. We show that the complexity of the diagram is O(na(m)+ str(P)), where str(P) is a parameter reflecting the number of straddles between pairs of clusters, which is O(m(n-m)). The bound reduces to O(n + str(P)) if the clusters are pairwise non-crossing. We also present a lower bound, establishing that the complexity of the FCVD can be O(n+m^2), even if the clusters have pairwise disjoint convex hulls. Our algorithm runs in O((n + str(P))log^3n)-time, and in certain special cases in O(n log(n)) time.Apreliminary and, partial portion of this work was presented at the 14th Latin American Theoretical Informatics Symposium (LATIN 2020) [23]. I. M., E. P., and Z.W were supported in part by the Swiss National Science Foundation, projects SNF 200021E-154387, SNF 200021E-201356. R. S. was supported by project PID2023-150725NB-I00/ MICIU/ AEI/ 10.13039/501100011033.Peer Reviewed20252025-06-0620252025-07-07journal articlehttp://purl.org/coar/resource_type/c_6501VoRhttp://purl.org/coar/version/c_970fb48d4fbd8a85info:eu-repo/semantics/articleapplication/pdfhttps://hdl.handle.net/2117/433637https://dx.doi.org/10.1007/s00453-025-01311-1reponame:UPCommons. Portal del coneixement obert de la UPCinstname:Universitat Politècnica de Catalunya (UPC)InglésengAgencia Estatal de Investigación http://doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2021-2023 PID2023-150725NB-I00 GRAFOS GEOMETRICOS Y ABSTRACTOS: TEORIA Y APLICACIONESopen accesshttp://purl.org/coar/access_right/c_abf2Attribution 4.0 Internationalhttp://creativecommons.org/licenses/by/4.0/info:eu-repo/semantics/openAccessoai:upcommons.upc.edu:2117/4336372026-05-27T15:37:01Z
dc.title.none.fl_str_mv The farthest color voronoi diagram in the plane
title The farthest color voronoi diagram in the plane
spellingShingle The farthest color voronoi diagram in the plane
Mantas, Ioannis
Farthest-site Voronoi diagram
Color Voronoi diagram
Point clusters
Color spanning disk
Straddles
Divide and conquer
Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi matemàtica
title_short The farthest color voronoi diagram in the plane
title_full The farthest color voronoi diagram in the plane
title_fullStr The farthest color voronoi diagram in the plane
title_full_unstemmed The farthest color voronoi diagram in the plane
title_sort The farthest color voronoi diagram in the plane
dc.creator.none.fl_str_mv Mantas, Ioannis
Papadopoulou, Evanthia|||0000-0003-0144-7384
Silveira, Rodrigo Ignacio|||0000-0003-0202-4543
Wang, Zeyu|||0009-0004-4207-198X
author Mantas, Ioannis
author_facet Mantas, Ioannis
Papadopoulou, Evanthia|||0000-0003-0144-7384
Silveira, Rodrigo Ignacio|||0000-0003-0202-4543
Wang, Zeyu|||0009-0004-4207-198X
author_role author
author2 Papadopoulou, Evanthia|||0000-0003-0144-7384
Silveira, Rodrigo Ignacio|||0000-0003-0202-4543
Wang, Zeyu|||0009-0004-4207-198X
author2_role author
author
author
dc.subject.none.fl_str_mv Farthest-site Voronoi diagram
Color Voronoi diagram
Point clusters
Color spanning disk
Straddles
Divide and conquer
Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi matemàtica
topic Farthest-site Voronoi diagram
Color Voronoi diagram
Point clusters
Color spanning disk
Straddles
Divide and conquer
Àrees temàtiques de la UPC::Matemàtiques i estadística::Anàlisi matemàtica
description The farthest-color Voronoi diagram (FCVD) is defined on a set of n points in the plane, where each point is labeled with one of m colors. The colored points constitute a family P of m clusters (sets) of points in the plane whose farthest-site Voronoi diagram is the FCVD. The diagram finds applications in problems related to facility location, shape matching, data imprecision, and others. In this paper we present structural properties of the FCVD, refine its combinatorial complexity bounds, and present efficient algorithms for its construction. We show that the complexity of the diagram is O(na(m)+ str(P)), where str(P) is a parameter reflecting the number of straddles between pairs of clusters, which is O(m(n-m)). The bound reduces to O(n + str(P)) if the clusters are pairwise non-crossing. We also present a lower bound, establishing that the complexity of the FCVD can be O(n+m^2), even if the clusters have pairwise disjoint convex hulls. Our algorithm runs in O((n + str(P))log^3n)-time, and in certain special cases in O(n log(n)) time.
publishDate 2025
dc.date.none.fl_str_mv 2025
2025-06-06
2025
2025-07-07
dc.type.none.fl_str_mv journal article
http://purl.org/coar/resource_type/c_6501
VoR
http://purl.org/coar/version/c_970fb48d4fbd8a85
dc.type.openaire.fl_str_mv info:eu-repo/semantics/article
format article
dc.identifier.none.fl_str_mv https://hdl.handle.net/2117/433637
https://dx.doi.org/10.1007/s00453-025-01311-1
url https://hdl.handle.net/2117/433637
https://dx.doi.org/10.1007/s00453-025-01311-1
dc.language.none.fl_str_mv Inglés
eng
language_invalid_str_mv Inglés
language eng
dc.relation.none.fl_str_mv Agencia Estatal de Investigación http://doi.org/10.13039/501100011033 Plan Estatal de Investigación Científica y Técnica y de Innovación 2021-2023 PID2023-150725NB-I00 GRAFOS GEOMETRICOS Y ABSTRACTOS: TEORIA Y APLICACIONES
dc.rights.none.fl_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
dc.rights.openaire.fl_str_mv info:eu-repo/semantics/openAccess
rights_invalid_str_mv open access
http://purl.org/coar/access_right/c_abf2
Attribution 4.0 International
http://creativecommons.org/licenses/by/4.0/
eu_rights_str_mv openAccess
dc.format.none.fl_str_mv application/pdf
dc.source.none.fl_str_mv reponame:UPCommons. Portal del coneixement obert de la UPC
instname:Universitat Politècnica de Catalunya (UPC)
instname_str Universitat Politècnica de Catalunya (UPC)
reponame_str UPCommons. Portal del coneixement obert de la UPC
collection UPCommons. Portal del coneixement obert de la UPC
repository.name.fl_str_mv
repository.mail.fl_str_mv
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