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 log² n) time, and reports in O(α log² n) 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.

Research supported in part by a Graduate School Faculty Summer Research Fellowship and by the Army High Performance Computing Research Center, both at the University of Minnesota.