Kinetic and Dynamic Data Structures for Convex Hulls and Upper Envelopes

Abstract

Let S be a set of n moving points in the plane. We present a kinetic and dynamic (randomized) data structure for maintaining the convex hull of S. The structure uses O(n) space, and processes an expected number of O(n 2 β s + 2(n)log n) critical events, each in O(log2 n) expected time, including O(n) insertions, deletions, and changes in the flight plans of the points. Here s is the maximum number of times where any specific triple of points can become collinear, β s (q)=λ s (q) / q, and λ s (q) is the maximum length of Davenport-Schinzel sequences of order s on n symbols. Compared with the previous solution of Basch et al.[2], our structure uses simpler certificates, uses roughly the same resources, and is also dynamic.