On maintaining the width and diameter of a planar point-set online

Abstract

Efficient online algorithms are presented for maintaining the (almost-exact) width and diameter of a dynamic planar point-set, S. Let n be the number of points currently in S, let W and D denote the width and diameter of S, respectively, and let α and Β be positive, integer-valued parameters. The algorithm for the width problem uses O(αn) space, supports updates in O(α log2n) time, and reports in O(α log2n) time an approximation, ŵ, to the width such that \(\hat W/W \leqslant \sqrt {1 + \tan ^2 \tfrac{\pi }{{4\alpha }}}\). The algorithm for the diameter problem uses O(Βn) space, supports updates in O(Βlogn) time, and reports in O(Β) time an approximation, D, to the diameter such that \(\hat D/D \geqslant \sin \left( {\tfrac{\beta }{{\beta + 1}}\tfrac{\pi }{2}} \right)\). Thus, for instance, even for α as small as 5, ŵ/W≤1.01, and for Β as small as 11, D/D≥.99. All bounds stated are worst-case. Both algorithms, but especially the one for the diameter problem, use well-understood data structures and should be simple to implement. The diameter result yields a fast implementation of the greedy heuristic for maximum-weight Euclidean matching and an efficient online algorithm to maintain approximate convex hulls in the plane.