Intersections with random geometric objects

Abstract We present a systematic study of the expected complexity of the intersection of geometric objects. We first study the expected size of the intersection between a random Voronoi diagram and a generic geometric object that consists of a finite collection of line segments in the plane. Using this result, we explore the intersection complexity of a random Voronoi diagram with the following target objects which may or may not be random: a line segment, a Voronoi diagram, a minimum spanning tree, a Gabriel graph, a relative neighborhood graph, a Hamiltonian circuit, a furthest point Voronoi diagram, a convex hull, a k-dimensional tree, and a rectangular grid.

[1]  Godfried T. Toussaint,et al.  PATTERN RECOGNITION AND GEOMETRICAL COMPLEXITY. , 1980 .

[2]  R. Sokal,et al.  A New Statistical Approach to Geographic Variation Analysis , 1969 .

[3]  D. Matula,et al.  Properties of Gabriel Graphs Relevant to Geographic Variation Research and the Clustering of Points in the Plane , 2010 .

[4]  Atsuyuki Okabe,et al.  Spatial Tessellations: Concepts and Applications of Voronoi Diagrams , 1992, Wiley Series in Probability and Mathematical Statistics.

[5]  Emo Welzl,et al.  On Spanning Trees with Low Crossing Numbers , 1992, Data Structures and Efficient Algorithms.

[6]  Luc Devroye,et al.  Intersections of random line segments , 1994, Int. J. Comput. Geom. Appl..

[7]  Godfried T. Toussaint,et al.  The relative neighbourhood graph of a finite planar set , 1980, Pattern Recognit..

[8]  Emo Welzl,et al.  Partition trees for triangle counting and other range searching problems , 1988, SCG '88.

[9]  Timothy M. Chan.  A Simple Trapezoid Sweep Algorithm for Reporting Red/Blue Segment Intersections , 1994, CCCG.

[10]  A. Rényi,et al.  über die konvexe Hülle von n zufällig gewählten Punkten , 1963 .

[11]  MatoušekJiří.  Geometric range searching , 1994 .

[12]  D. Kirkpatrick,et al.  A Framework for Computational Morphology , 1985 .

[13]  Hanan Samet,et al.  Applications of spatial data structures , 1989 .

[14]  Jon Louis Bentley,et al.  Multidimensional binary search trees used for associative searching , 1975, CACM.

[15]  Jirí Matousek,et al.  Geometric range searching , 1994, CSUR.

[16]  Thomas Ertl,et al.  Computer Graphics - Principles and Practice, 3rd Edition , 2014 .

[17]  Luc Devroye,et al.  Lecture Notes on Bucket Algorithms , 1986, Progress in Computer Science.

[18]  J. Møller.  Random tessellations in ℝ d , 1989, Advances in Applied Probability.

[19]  M. Carter.  Computer graphics: Principles and practice , 1997 .

[20]  Luc Devroye,et al.  Coupled Samples in Simulation , 1990, Oper. Res..

[21]  L. Devroye.  THE EXPECTED SIZE OF SOME GRAPHS IN COMPUTATIONAL GEOMETRY , 1988 .

[22]  Philippe Flajolet,et al.  Partial match retrieval of multidimensional data , 1986, JACM.

[23]  C. Yap.  Exact computational geometry and tolerancing metrology , 1994 .

[24]  L. Few.  The shortest path and the shortest road through n points , 1955 .

[25]  F. Frances Yao,et al.  Computational Geometry , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[26]  Shlomo Moran,et al.  On the length of optimal TSP circuits in sets of bounded diameter , 1984, J. Comb. Theory, Ser. B.

[27]  Utpal Roy,et al.  Establishment of a pair of concentric circles with the minimum radial separation for assessing roundness error , 1992, Comput. Aided Des..