Computing relative neighbourhood graphs in the plane

Abstract The relative neighbourhood graph (RNG) of a set of N points connects the relative neighbours, i.e. a pair of points is connected by an edge if those points are at least as close to each other as to any other point. The paper presents two new algorithms for constructing RNG in two-dimensional Euclidean space. The method is to determine a supergraph for RNG which can then be thinned efficiently from the extra edges. The first algorithm is simple, and works in O ( N 2 ) time. The worst case running time of the second algorithm is also O ( N 2 ), but its average case running time is O ( N ) for points from a homogeneous planar Poisson point process. Experimental tests have shown the usefulness of the approach.

[1]  Joseph O'Rourke.  Computing the relative neighborhood graph in the L1 and Linfinity metrics , 1982, Pattern Recognit..

[2]  Kenneth J. Supowit,et al.  The Relative Neighborhood Graph, with an Application to Minimum Spanning Trees , 1983, JACM.

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

[4]  Richard C. T. Lee,et al.  The average performance analysis of a closest‐pair algorithm , 1984 .

[5]  Bruce W. Weide,et al.  Optimal Expected-Time Algorithms for Closest Point Problems , 1980, TOMS.

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

[7]  Godfried T. Toussaint,et al.  Comment: Algorithms for computing relative neighbourhood graph , 1980 .

[8]  R. Urquhart.  Algorithms for computation of relative neighbourhood graph , 1980 .

[9]  Robert E. Tarjan,et al.  Finding Minimum Spanning Trees , 1976, SIAM J. Comput..

[10]  Arne Maus,et al.  Delaunay triangulation and the convex hull ofn points in expected linear time , 1984, BIT.

[11]  R. E. Miles.  On the homogeneous planar Poisson point process , 1970 .

[12]  B. Weide.  Statistical methods in algorithm design and analysis. , 1978 .

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

[14]  Godfried T. Toussaint.  ALGORITHMS FOR COMPUTING RELATIVE NEIGHBOURHOOD GRAPH. , 1980 .

[15]  Richard C. T. Lee,et al.  On the average length of Delaunay triangulations , 1984, BIT.

[16]  S. Chern.  Review: Luis A. Santaló, Integral geometry and geometric probability , 1977 .

[17]  L. Santaló.  Integral geometry and geometric probability , 1976 .

[18]  Andrew Chi-Chih Yao,et al.  On Constructing Minimum Spanning Trees in k-Dimensional Spaces and Related Problems , 1977, SIAM J. Comput..