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...
| Autores: | , , , |
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| 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 |
| Sumario: | 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. |
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