The upper envelope of Voronoi surfaces and its applications

Daniel P. Huttenlocher; Klara Kedem; Micha Sharir

doi:10.1145/109648.109670

The upper envelope of Voronoi surfaces and its applications

Daniel P. Huttenlocher, Klara Kedem, Micha Sharir

School of Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

31 Scopus citations

Abstract

Given a set S of sources (points or segments), we consider the surface that is the graph of the function d(x) - min_p∈s ρ(x,p), for some metric ρ. This surface is closely related to the Voronoi diagram, Vor(S), of S under the metric ρ. The upper envelope of a set of these Voronoi surfaces, each defined for a different set of sources, can be used to solve a number of problems, including finding the minimum Hausdorff distance between two sets of points or segments under translation, and determining the optimal placement of a site with respect to sets of utilities. We derive bounds on the number of vertices on the upper envelope of m Voronoi surfaces, provide efficient algorithms to calculate these vertices, and discuss applications to the aforementioned problems.

Original language	English
Title of host publication	Proceedings of the Annual Symposium on Computational Geometry
Publisher	Association for Computing Machinery
Pages	194-203
Number of pages	10
ISBN (Print)	0897914260
DOIs	https://doi.org/10.1145/109648.109670
State	Published - 1 Jun 1991
Event	7th Annual Symposium on Computational Geometry, SCG 1991 - North Conway, United States Duration: 10 Jun 1991 → 12 Jun 1991

Publication series

Name	Proceedings of the Annual Symposium on Computational Geometry

Conference

Conference	7th Annual Symposium on Computational Geometry, SCG 1991
Country/Territory	United States
City	North Conway
Period	10/06/91 → 12/06/91

Funding

Funders	Funder number
Israeli Academy of Sciences
Kodak Corporation
Office of Navaf Research
Pikkowski-Vrdazzi Fund	04601-90
U.S.-Israeli Binational Science Foundation
National Science Foundation	CCR-89-01484, IRI-9057928

Access to Document

10.1145/109648.109670

Cite this

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title = "The upper envelope of Voronoi surfaces and its applications",

abstract = "Given a set S of sources (points or segments), we consider the surface that is the graph of the function d(x) - minp∈s ρ(x,p), for some metric ρ. This surface is closely related to the Voronoi diagram, Vor(S), of S under the metric ρ. The upper envelope of a set of these Voronoi surfaces, each defined for a different set of sources, can be used to solve a number of problems, including finding the minimum Hausdorff distance between two sets of points or segments under translation, and determining the optimal placement of a site with respect to sets of utilities. We derive bounds on the number of vertices on the upper envelope of m Voronoi surfaces, provide efficient algorithms to calculate these vertices, and discuss applications to the aforementioned problems.",

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Huttenlocher, DP, Kedem, K & Sharir, M 1991, The upper envelope of Voronoi surfaces and its applications. in Proceedings of the Annual Symposium on Computational Geometry. Proceedings of the Annual Symposium on Computational Geometry, Association for Computing Machinery, pp. 194-203, 7th Annual Symposium on Computational Geometry, SCG 1991, North Conway, United States, 10/06/91. https://doi.org/10.1145/109648.109670

The upper envelope of Voronoi surfaces and its applications. / Huttenlocher, Daniel P.; Kedem, Klara; Sharir, Micha.
Proceedings of the Annual Symposium on Computational Geometry. Association for Computing Machinery, 1991. p. 194-203 (Proceedings of the Annual Symposium on Computational Geometry).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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The upper envelope of Voronoi surfaces and its applications

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Conference

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