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( ⁿ²βs+ ₂(n)logn) critical events, each in O( ^log2n) 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, ^βs(q) = ^λs(q)/q, and ^λs(q) is the maximum length of Davenport-Schinzel sequences of order s on n symbols. Compared with the previous solution of Basch, Guibas and Hershberger [J. Basch, L.J. Guibas, J. Hershberger, Data structures for mobile data, J. Algorithms 31 (1999) 1-28], our structure uses simpler certificates, uses roughly the same resources, and is also dynamic.