On Topological Changes in the Delaunay Triangulation of Moving Points

Abstract

Let P be a collection of n points moving along pseudo-algebraic trajectories in the plane. (So, in particular, there are constants s, c > 0 such that any four points are co-circular at most s times, and any three points are collinear at most c times.) One of the hardest open problems in combinatorial and computational geometry is to obtain a nearly quadratic upper bound, or at least a sub-cubic bound, on the maximum number of discrete changes that the Delaunay triangulation DT(P) of P experiences during the motion of the points of P. In this paper, we obtain an upper bound of O(n 2+ε), for any ε > 0, under the assumptions that (i) any four points can be co-circular at most twice and (ii) either no triple of points can be collinear more than twice or no ordered triple of points can be collinear more than once.

On Kinetic Delaunay Triangulations

“…In this section we introduce the notion of a Delaunay crossing, which plays a central role both in this article and in its predecessor [Rubin 2013], and express this quantity N(n) in terms of the maximum numbers of Delaunay crossings that can arise in smaller sets of moving points.…”

“…In Section 3, we show that the number of Delaunay co-circularities is dominated by the maximum possible number of Delaunay crossings. Notice the previously sketched argument (which appears in Rubin [2013]) works only for the first and the last Delaunay co-circularities of the quadruple.…”

“…The following useful result on A pq , which is one of the major tools in our analysis, was established in Rubin [2013] by applying routine techniques for analyzing planar arrangements. For the sake of completeness, we provide its proof in Appendix C.…”

“…The Red-Blue Arrangement. As in Guibas et al [1992] and Rubin [2013], we use the so-called red-blue arrangement to facilitate the analysis of co-circularities whose corresponding discs touch the same two points p, q ∈ P. For the sake of completeness, we provide here a formal definition of this arrangement.…”

Kinetic data structures for all nearest neighbors and closest pair in the plane

“…Kinetic versions of problems have been studied extensively over the past decade, e.g., kinetic Delaunay triangulation [6,22], kinetic Yao graph [2], kinetic point-set embeddability [20], kinetic Euclidean minimum spanning tree [2,21], kinetic closest pair [5,8], kinetic convex hull [7], kinetic spanners [1,16], and kinetic range searching [3].…”