Computing farthest neighbors on a convex polytope

Otfried Cheong; Chan Su Shin; Antoine Vigneron

doi:10.1007/3-540-44679-6_18

Computing farthest neighbors on a convex polytope

Otfried Cheong, Chan Su Shin, Antoine Vigneron

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

1 Scopus citations

Abstract

Let N be a set of n points in convex position in ℝ³. The farthest-point Voronoi diagram of N partitions ℝ³ into n convex cells. We consider the intersection G(N) of the diagram with the boundary of the convex hull of N. We give an algorithm that computes an implicit representation of G(N) in expected O(n log² n) time. More precisely, we compute the combinatorial structure of G(N), the coordinates of its vertices, and the equation of the plane defining each edge of G(N). The algorithm allows us to solve the all-pairs farthest neighbor problem for N in expected time O(n log² n), and to perform farthest-neighbor queries on N in O(log² n) time with high probability. This can be applied to find a Euclidean maximum spanning tree and a diameter 2-clustering of N in expected O(n log⁴ n) time.

Original language	English (US)
Title of host publication	Computing and Combinatorics - 7th Annual International Conference, COCOON 2001, Proceedings
Editors	Jie Wang
Publisher	Springer Verlag
Pages	159-169
Number of pages	11
ISBN (Print)	9783540424949
DOIs	https://doi.org/10.1007/3-540-44679-6_18
State	Published - 2001
Externally published	Yes
Event	7th Annual International Conference on Computing and Combinatorics, COCOON 2001 - Guilin, China Duration: Aug 20 2001 → Aug 23 2001

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	2108
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Other

Other	7th Annual International Conference on Computing and Combinatorics, COCOON 2001
Country/Territory	China
City	Guilin
Period	08/20/01 → 08/23/01

Bibliographical note

Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2001.

ASJC Scopus subject areas

Theoretical Computer Science
General Computer Science

Access to Document

10.1007/3-540-44679-6_18

Cite this

Cheong, O., Shin, C. S., & Vigneron, A. (2001). Computing farthest neighbors on a convex polytope. In J. Wang (Ed.), Computing and Combinatorics - 7th Annual International Conference, COCOON 2001, Proceedings (pp. 159-169). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2108). Springer Verlag. https://doi.org/10.1007/3-540-44679-6_18

Cheong, Otfried ; Shin, Chan Su ; Vigneron, Antoine. / Computing farthest neighbors on a convex polytope. Computing and Combinatorics - 7th Annual International Conference, COCOON 2001, Proceedings. editor / Jie Wang. Springer Verlag, 2001. pp. 159-169 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)).

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title = "Computing farthest neighbors on a convex polytope",

abstract = "Let N be a set of n points in convex position in ℝ3. The farthest-point Voronoi diagram of N partitions ℝ3 into n convex cells. We consider the intersection G(N) of the diagram with the boundary of the convex hull of N. We give an algorithm that computes an implicit representation of G(N) in expected O(n log2 n) time. More precisely, we compute the combinatorial structure of G(N), the coordinates of its vertices, and the equation of the plane defining each edge of G(N). The algorithm allows us to solve the all-pairs farthest neighbor problem for N in expected time O(n log2 n), and to perform farthest-neighbor queries on N in O(log2 n) time with high probability. This can be applied to find a Euclidean maximum spanning tree and a diameter 2-clustering of N in expected O(n log4 n) time.",

author = "Otfried Cheong and Shin, {Chan Su} and Antoine Vigneron",

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doi = "10.1007/3-540-44679-6_18",

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Cheong, O, Shin, CS & Vigneron, A 2001, Computing farthest neighbors on a convex polytope. in J Wang (ed.), Computing and Combinatorics - 7th Annual International Conference, COCOON 2001, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 2108, Springer Verlag, pp. 159-169, 7th Annual International Conference on Computing and Combinatorics, COCOON 2001, Guilin, China, 08/20/01. https://doi.org/10.1007/3-540-44679-6_18

Computing farthest neighbors on a convex polytope. / Cheong, Otfried; Shin, Chan Su; Vigneron, Antoine.
Computing and Combinatorics - 7th Annual International Conference, COCOON 2001, Proceedings. ed. / Jie Wang. Springer Verlag, 2001. p. 159-169 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 2108).

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

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N2 - Let N be a set of n points in convex position in ℝ3. The farthest-point Voronoi diagram of N partitions ℝ3 into n convex cells. We consider the intersection G(N) of the diagram with the boundary of the convex hull of N. We give an algorithm that computes an implicit representation of G(N) in expected O(n log2 n) time. More precisely, we compute the combinatorial structure of G(N), the coordinates of its vertices, and the equation of the plane defining each edge of G(N). The algorithm allows us to solve the all-pairs farthest neighbor problem for N in expected time O(n log2 n), and to perform farthest-neighbor queries on N in O(log2 n) time with high probability. This can be applied to find a Euclidean maximum spanning tree and a diameter 2-clustering of N in expected O(n log4 n) time.

AB - Let N be a set of n points in convex position in ℝ3. The farthest-point Voronoi diagram of N partitions ℝ3 into n convex cells. We consider the intersection G(N) of the diagram with the boundary of the convex hull of N. We give an algorithm that computes an implicit representation of G(N) in expected O(n log2 n) time. More precisely, we compute the combinatorial structure of G(N), the coordinates of its vertices, and the equation of the plane defining each edge of G(N). The algorithm allows us to solve the all-pairs farthest neighbor problem for N in expected time O(n log2 n), and to perform farthest-neighbor queries on N in O(log2 n) time with high probability. This can be applied to find a Euclidean maximum spanning tree and a diameter 2-clustering of N in expected O(n log4 n) time.

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Cheong O, Shin CS, Vigneron A. Computing farthest neighbors on a convex polytope. In Wang J, editor, Computing and Combinatorics - 7th Annual International Conference, COCOON 2001, Proceedings. Springer Verlag. 2001. p. 159-169. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/3-540-44679-6_18

Computing farthest neighbors on a convex polytope

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