Computing farthest neighbors on a convex polytope

Authors:
Otfried Cheong;Chan-Su Shin;Antoine Vigneron
Affiliations:
Institute of Information and Computing Sciences, Utrecht University, Utrecht, Netherlands;School of Electronic and Information Engineering, Hankuk University of Foreign Studies, 449-791, Wangsan-ri, Mohyeon-myeon, Yongin-Shi, Gyeonggi-do, South Korea;Department of Computer Science, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China
Venue:
Theoretical Computer Science - Computing and combinatorics
Year:
2003

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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.