On Kinetic Delaunay Triangulations: A Near Quadratic Bound for Unit Speed Motions

Abstract

Let $P$ be a collection of $n$ points in the plane, each moving along some straight line at unit speed. We obtain an almost tight upper bound of $O(n^{2+ε})$, for any $ε>0$, on the maximum number of discrete changes that the Delaunay triangulation $\mathbb{DT}(P)$ of $P$ experiences during this motion. Our analysis is cast in a purely topological setting, where we only assume that (i) any four points can be co-circular at most three times, and (ii) no triple of points can be collinear more than twice; these assumptions hold for unit speed motions.

138 pages+ Appendix of 7 pages. A preliminary version has appeared in Proceedings of the 54th Annual Symposium on Foundations of Computer Science (FOCS 2013). The paper extends the result of http://arxiv.org/abs/1304.3671 to more general motions. The presentation is self-contained with main ideas delivered in Sections 1--4