Computing Farthest Neighbors on a Convex Polytope

Abstract

Let N be a set of n points in convex position in ∝³. The farthest-point Voronoi diagram of N partitions ∝³ 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 de?ning 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.

This research was partially supported by the Hong Kong Research Grants Council.Part of it was done when the first two authors were at HKUST. The second author’s research was supported by grant No. 98-0102-07-01-3 from KOSEF and also supported by BK21 Research Professor Program at KAIST.