On the construction of abstract voronoi diagrams

We show that the abstract Voronoi diagram ofn sites in the plane can be constructed in timeO(n logn) by a randomized algorithm. This yields an alternative, but simpler,O(n logn) algorithm in many previously considered cases and the firstO(n logn) algorithm in some cases, e.g., disjoint convex sites with the Euclidean distance function. Abstract Voronoi diagrams are given by a family of bisecting curves and were recently introduced by Klein [13]. Our algorithm is based on Clarkson and Shor's randomized incremental construction technique [7].

[1]  Kevin Q. Brown,et al.  Voronoi Diagrams from Convex Hulls , 1979, Inf. Process. Lett..

[2]  Steven Fortune,et al.  A sweepline algorithm for Voronoi diagrams , 1986, SCG '86.

[3]  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..

[4]  Steven Fortune.  A Fast Algorithm for Polygon Containment by Translation (Extended Abstract) , 1985, ICALP.

[5]  Franz Aurenhammer,et al.  Improved Algorithms for Discs and Balls Using Power Diagrams , 1988, J. Algorithms.

[6]  Hiroshi Imai,et al.  Voronoi Diagram in the Laguerre Geometry and its Applications , 1985, SIAM J. Comput..

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

[8]  David G. Kirkpatrick,et al.  Efficient computation of continuous skeletons , 1979, 20th Annual Symposium on Foundations of Computer Science (sfcs 1979).

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

[10]  Joseph S. B. Mitchell,et al.  The weighted region problem , 1987, SCG '87.

[11]  Kurt Mehlhorn,et al.  On the Construction of Abstract Voronoi Diagrams, II , 1990, SIGAL International Symposium on Algorithms.

[12]  Kenneth L. Clarkson,et al.  Applications of random sampling in computational geometry, II , 1988, SCG '88.

[13]  Rolf Klein,et al.  Abstract Voronoi Diagrams and their Applications , 1988, Workshop on Computational Geometry.

[14]  Kenneth L. Clarkson,et al.  Algorithms for diametral pairs and convex hulls that are optimal, randomized, and incremental , 1988, SCG '88.

[15]  Rolf Klein,et al.  Voronoi Diagrams in the Moscow Metric (Extended Abstract) , 1987, WG.

[16]  Derick Wood,et al.  Voronoi Diagrams Based on General Metrics in the Plane , 1988, STACS.

[17]  Micha Sharir,et al.  Intersection and Closest-Pair Problems for a Set of Planar Discs , 1985, SIAM J. Comput..

[18]  Raimund Seidel,et al.  Voronoi diagrams and arrangements , 1986, Discret. Comput. Geom..

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

[20]  Franz Aurenhammer,et al.  An optimal algorithm for constructing the weighted voronoi diagram in the plane , 1984, Pattern Recognit..

[21]  Franz Aurenhammer,et al.  Power Diagrams: Properties, Algorithms and Applications , 1987, SIAM J. Comput..

[22]  Rolf Klein,et al.  Concrete and Abstract Voronoi Diagrams , 1990, Lecture Notes in Computer Science.