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: otfried@cs.uu.nl (O. Cheong), cssin@control.hufs.ac.kr (C.-S. Shin), antoine@cs.ust.hk (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

UR - http://www.scopus.com/inward/record.url?scp=0037265307&partnerID=8YFLogxK

U2 - 10.1016/S0304-3975(02)00431-0

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 -