3 research outputs found
Ready, Set, Go! The Voronoi Diagram of Moving Points that Start from a Line
It is an outstanding open problem of computational geometry to prove a nearquadratic upper bound on the number of combinatorial changes in the Voronoi diagram of points moving at a common constant speed along linear trajectories in the plane. In this note we observe that this quantity is Θ(n 2) if the points start their movement from a common line.
On Kinetic Delaunay Triangulations: A Near Quadratic Bound for Unit Speed Motions
Let be a collection of points in the plane, each moving along some
straight line at unit speed. We obtain an almost tight upper bound of
, for any , on the maximum number of discrete
changes that the Delaunay triangulation of 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.Comment: 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--