On Higher Order Voronoi Diagrams of Line Segments

We analyze structural properties of the order-k Voronoi diagram of line segments, which surprisingly has not received any attention in the computational geometry literature. We show that order-k Voronoi regions of line segments may be disconnected; in fact a single order-k Voronoi region may consist of Ω(n) disjoint faces. Nevertheless, the structural complexity of the order-k Voronoi diagram of non-intersecting segments remains O(k(n − k)) similarly to points. For intersecting line segments the structural complexity remains O(k(n − k)) for k ≥ n/2.

[1]  Franz Aurenhammer,et al.  Voronoi Diagrams , 2000, Handbook of Computational Geometry.

[2]  D. T. Lee,et al.  Generalization of Voronoi Diagrams in the Plane , 1981, SIAM J. Comput..

[3]  Michael Ian Shamos,et al.  Closest-point problems , 1975, 16th Annual Symposium on Foundations of Computer Science (sfcs 1975).

[4]  Evanthia Papadopoulou,et al.  Net-Aware Critical Area Extraction for Opens in VLSI Circuits Via Higher-Order Voronoi Diagrams , 2011, IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems.

[5]  Jean-Daniel Boissonnat,et al.  A semidynamic construction of higher-order voronoi diagrams and its randomized analysis , 1993, Algorithmica.

[6]  D. T. Lee,et al.  On k-Nearest Neighbor Voronoi Diagrams in the Plane , 1982, IEEE Transactions on Computers.

[7]  Franz Aurenhammer,et al.  Farthest line segment Voronoi diagrams , 2006, Inf. Process. Lett..

[8]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[9]  Arnold L. Rosenberg,et al.  Stabbing line segments , 1982, BIT.

[10]  Noga Alon,et al.  The number of small semispaces of a finite set of points in the plane , 1986, J. Comb. Theory, Ser. A.

[11]  Jirí Matousek,et al.  Constructing levels in arrangements and higher order Voronoi diagrams , 1994, SCG '94.

[12]  M. Karavelas.  A robust and efficient implementation for the segment Voronoi diagram , 2004 .

[13]  Chee-Keng Yap,et al.  AnO(n logn) algorithm for the voronoi diagram of a set of simple curve segments , 1987, Discret. Comput. Geom..

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

[15]  J. Sack,et al.  Handbook of computational geometry , 2000 .