TY - GEN

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).
Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2001.

PY - 2001

Y1 - 2001

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.

UR - http://www.scopus.com/inward/record.url?scp=0012794484&partnerID=8YFLogxK

U2 - 10.1007/3-540-44679-6_18

DO - 10.1007/3-540-44679-6_18

M3 - Conference contribution

AN - SCOPUS:0012794484

SN - 9783540424949

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 159

EP - 169

BT - Computing and Combinatorics - 7th Annual International Conference, COCOON 2001, Proceedings

A2 - Wang, Jie

PB - Springer Verlag

T2 - 7th Annual International Conference on Computing and Combinatorics, COCOON 2001

Y2 - 20 August 2001 through 23 August 2001

ER -