Kinetic and Dynamic Data Structures for Convex Hulls and Upper Envelopes

摘要

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(log~2 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., our structure uses simpler certificates, uses roughly the same resources, and is also dynamic.