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 ( 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, 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.