Computing farthest neighbors on a convex polytope

Otfried Cheong; Chan Su Shin; Antoine Vigneron

doi:10.1016/S0304-3975(02)00431-0

Computing farthest neighbors on a convex polytope

Otfried Cheong^*, Chan Su Shin, Antoine Vigneron

^*Corresponding author for this work

Research output: Contribution to journal › Conference article › peer-review

15 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(nlog²n) 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(nlog²n), and to perform farthest-neighbor queries on N in O(log²n) time with high probability.

Original language	English (US)
Pages (from-to)	47-58
Number of pages	12
Journal	Theoretical Computer Science
Volume	296
Issue number	1
DOIs	https://doi.org/10.1016/S0304-3975(02)00431-0
State	Published - 2003
Externally published	Yes
Event	Computing and Combinatorics - Guilin, China Duration: Aug 20 2001 → Aug 23 2001

Bibliographical note

Funding Information:
This research was partially supported by the Hong Kong Research Grants Council and partially by grant No. R05-2002-000-00780-0 from the Korea Science & Engineering Foundation. Part of it was done when the 1rst two authors were at HKUST. ∗Corresponding author. E-mail addresses: [email protected] (O. Cheong), [email protected] (C.-S. Shin), [email protected] (A. Vigneron).

Keywords

3D
Computational geometry
Farthest neighbors
Farthest-point Voronoi Diagram
Polytope

ASJC Scopus subject areas

Theoretical Computer Science
General Computer Science

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10.1016/S0304-3975(02)00431-0

Cite this

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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(nlog2n) 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(nlog2n), and to perform farthest-neighbor queries on N in O(log2n) time with high probability.",

keywords = "3D, Computational geometry, Farthest neighbors, Farthest-point Voronoi Diagram, Polytope",

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

note = "Funding Information: This research was partially supported by the Hong Kong Research Grants Council and partially by grant No. R05-2002-000-00780-0 from the Korea Science & Engineering Foundation. Part of it was done when the 1rst two authors were at HKUST. ∗Corresponding author. E-mail addresses: [email protected] (O. Cheong), [email protected] (C.-S. Shin), [email protected] (A. Vigneron).; Computing and Combinatorics ; Conference date: 20-08-2001 Through 23-08-2001",

year = "2003",

doi = "10.1016/S0304-3975(02)00431-0",

language = "English (US)",

volume = "296",

pages = "47--58",

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TY - JOUR

T1 - Computing farthest neighbors on a convex polytope

AU - Cheong, Otfried

AU - Shin, Chan Su

AU - Vigneron, Antoine

N1 - Funding Information: This research was partially supported by the Hong Kong Research Grants Council and partially by grant No. R05-2002-000-00780-0 from the Korea Science & Engineering Foundation. Part of it was done when the 1rst two authors were at HKUST. ∗Corresponding author. E-mail addresses: [email protected] (O. Cheong), [email protected] (C.-S. Shin), [email protected] (A. Vigneron).

PY - 2003

Y1 - 2003

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(nlog2n) 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(nlog2n), and to perform farthest-neighbor queries on N in O(log2n) time with high probability.

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(nlog2n) 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(nlog2n), and to perform farthest-neighbor queries on N in O(log2n) time with high probability.

KW - 3D

KW - Computational geometry

KW - Farthest neighbors

KW - Farthest-point Voronoi Diagram

KW - Polytope

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DO - 10.1016/S0304-3975(02)00431-0

M3 - Conference article

AN - SCOPUS:0037265307

SN - 0304-3975

VL - 296

SP - 47

EP - 58

JO - Theoretical Computer Science

JF - Theoretical Computer Science

IS - 1

T2 - Computing and Combinatorics

Y2 - 20 August 2001 through 23 August 2001

ER -

Computing farthest neighbors on a convex polytope

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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