Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions

Abstract

Let P be a set of n points and Q a convex k-gon in R 2 . We analyze in detail the topological (or discrete) changes in the structure of the Voronoi diagram and the Delaunay triangulation of P, under the convex distance function defined by Q, as the points of P move along prespecified continuous trajectories. Assuming that each point of P moves along an algebraic trajectory of bounded degree, we establish an upper bound of O(k 4 nλ r (n)) on the number of topological changes experienced by the diagrams throughout the motion; here λ r (n) is the maximum length of an (n, r)-Davenport-Schinzel sequence, and r is a constant depending on the algebraic degree of the motion of the points. Finally, we describe an algorithm for efficiently maintaining the above structures, using the kinetic data structure (KDS) framework.

Stable Delaunay Graphs

“…More precisely, we say that the distance function induced by a compact convex set Q is α-close to the Euclidean norm if Q is contained in the unit disk D O and contains the disk D I = (cos α)D O both centered in the origin. 3 See Figure 3. In particular, for k = π/α, the regular k-gon Q k is such a set, as easy trigonometry shows.…”

“…In particular, if Q is a regular k-gon for k ≥ 2π/α, then the above theorem holds for Q. In the companion paper [3], we have presented an efficient kinetic data structure for maintaining the Delaunay triangulation and Voronoi diagram of P under a polygonal convex distance function. Using this result, we obtain the second main result of the paper: Theorem 1.2.…”

“…Chew [11] showed that the Delaunay triangulation of P under the L 1 -or L ∞ -metric experiences only a near-quadratic number of events, if the motion of the points of P is algebraic of bounded degree. In the companion paper [3], we present a kinetic data structure for maintaining the Voronoi diagram and Delaunay triangulation of P under a polygonal convex distance function for an arbitrary convex polygon Q. (see Section 2 for the definition) that processes only a near-quadratic number of events, and can be updated in O(log n) time at each event.…”

“…and then VD Q (P) and DT Q (P) are defined in complete analogy to the Euclidean case. We refer the reader to [3] for formal definitions and details of these structures. Throughout this paper, we will drop the superscript Q from Vor Q , VD Q , DT Q when referring to them under the Euclidean norm.…”

Delaunay Triangulation

“…A Delaunay Triangulation has a dual graph called a Voronoi diagram, which is formed by using circumcenters of Delaunay triangles (or tetrahedra), thus some authors have taken advantage of this to explore both geometrical structures (Watson 1981;Chew 1990;Agarwal et al 2015;Allen et al 2016). The most common methods to construct a Delaunay Triangulation are Lawson method (Lawson 1977), Bowyer method (Bowyer 1981) and Watson method (Watson 1981).…”