Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions

Let P be a set of n points and Q a convex k-gon in $${\mathbb {R}}^2$$R2. 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 $$O(k^4n\lambda _r(n))$$O(k4nλr(n)) on the number of topological changes experienced by the diagrams throughout the motion; here $$\lambda _r(n)$$λr(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.

[1]  Robert L. Scot Drysdale,et al.  Voronoi diagrams based on convex distance functions , 1985, SCG '85.

[2]  Steven Fortune,et al.  Voronoi Diagrams and Delaunay Triangulations , 2004, Handbook of Discrete and Computational Geometry, 2nd Ed..

[3]  Frank K. Hwang,et al.  An O(n log n) Algorithm for Rectilinear Minimal Spanning Trees , 1979, JACM.

[4]  Chak-Kuen Wong,et al.  Voronoi Diagrams in L1 (Linfty) Metrics with 2-Dimensional Storage Applications , 1980, SIAM J. Comput..

[5]  Richard C. T. Lee,et al.  Voronoi Diagrams of Moving Points in the Plane , 1990, FSTTCS.

[6]  A. Stillings Modeling Motion , 2005 .

[7]  Leonidas J. Guibas,et al.  Data structures for mobile data , 1997, SODA '97.

[8]  Natan Rubin,et al.  On Topological Changes in the Delaunay Triangulation of Moving Points , 2012, SoCG '12.

[9]  Micha Sharir,et al.  On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles , 1986, Discret. Comput. Geom..

[10]  D. T. Lee,et al.  Two-Dimensional Voronoi Diagrams in the Lp-Metric , 1980, J. ACM.

[11]  Mariette Yvinec,et al.  Voronoi Diagrams in Higher Dimensions under Certain Polyhedral Distance Functions , 1995, SCG '95.

[12]  Rolf Klein,et al.  Convex distance functions in 3-space are different , 1993, SCG '93.

[13]  Leonidas J. Guibas,et al.  Kinetic stable Delaunay graphs , 2010, SCG.

[14]  Dan Halperin,et al.  CGAL Arrangements and Their Applications - A Step-by-Step Guide , 2012, Geometry and Computing.

[15]  Richard C. T. Lee,et al.  Voronoi diagrams of moving points in the plane , 1990, Int. J. Comput. Geom. Appl..

[16]  Robert L. Scot Drysdale,et al.  A practical algorithm for computing the Delaunay triangulation for convex distance functions , 1990, SODA '90.

[17]  Leonidas J. Guibas,et al.  Stable Delaunay Graphs , 2015, Discret. Comput. Geom..

[18]  L. Paul Chew,et al.  Near-quadratic Bounds for the L1Voronoi Diagram of Moving Points , 1993, Comput. Geom..

[19]  Joseph O'Rourke,et al.  Handbook of Discrete and Computational Geometry, Second Edition , 1997 .

[20]  Chak-Kuen Wong,et al.  On Some Distance Problems in Fixed Orientations , 1987, SIAM J. Comput..

[21]  Micha Sharir,et al.  Planning a purely translational motion for a convex object in two-dimensional space using generalized Voronoi diagrams , 2016, Discret. Comput. Geom..

[22]  Natan Rubin,et al.  On Kinetic Delaunay Triangulations: A Near Quadratic Bound for Unit Speed Motions , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[23]  Chee-Keng Yap,et al.  A geometric consistency theorem for a symbolic perturbation scheme , 1988, SCG '88.

[24]  Sven Skyum,et al.  A Sweepline Algorithm for Generalized Delaunay Triangulations , 1991 .

[25]  Micha Sharir,et al.  Polyhedral Voronoi Diagrams of Polyhedra in Three Dimensions , 2002, SCG '02.

[26]  Lihong Ma,et al.  Bisectors and Voronoi Diagrams for Convex Distance Functions , 2000 .

[27]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

[28]  Micha Sharir,et al.  Davenport-Schinzel sequences and their geometric applications , 1995, Handbook of Computational Geometry.