The Symmetric All-furthest- Neighbor Problem

Given a set 9 of n points on the plane, a symmerric_furfhesr-neighbor (SFN) pair of points p, q is one such that both p and q are furthest from each other among the points in 8. A pair of points is untipodal if it admits parallel lines of support. In this paper it is shown that a SFN pair of 9 is both a set of extreme points of 9 and an antipodul pair of 8. It is also shown that an asymmerricfurthest-neighbor (ASFN) pair is not necessarily anripodul. Furthermore, if 9 is such that no two distances are equal, it is shown that as many as, and no more than, Ln/ZJ pairs of points are SFN pairs. A polygon is unimodul if for each vertex pk, k = 1, , n the distance function defined by the euclidean distance between pI and the remaining vertices (traversed in order) contains only one local maximum. The fastest existing algorithms for computing all the ASFN or SFN pairs of either a set of points, a simple polygon, or a convex polygon, require O(n logn) running time. It is shown that the above results lead to an O(n) algorithm for computing all the SFN pairs of vertices of a z&modal polygon.