Kinetic and dynamic data structures for convex hulls and upper envelopes

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 ²β_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. [2], our structure uses simpler certificates, uses roughly the same resources, and is also dynamic.