A linear time algorithm for finding all farthest neighbors in a convex polygon

Abstract

Aggarwal et al. [A. Aggarwal, M.M. Klawe, S. Moran, P. Shor and R. Wilber, Geometric applications of a matrix-searching algorithm, Algorithmica 2 (1987) 209–233] showed how to compute in O( n) time one farthest neighbor for every vertex of a convex n-gon. In this note we extend this result to obtain a linear time algorithm for finding all farthest neighbors for every vertex of a convex polygon. Our algorithm yields a linear time solution to the symmetric all-farthest neighbors problem for simple polygons, thereby settling an open question raised by Toussaint in 1983 [G.T. Toussaint, The symmetric all-farthest neighbor problem, Comp. Math. Appl. 9 (6) (1983) 747–753.]