Local properties of geometric graphs

We propose a definition of locality for properties of geometric graphs. We measure the local density of graphs using region-counting distances between pairs of vertices, and we use this density to define local properties of classes of graphs. We illustrate locality by introducing the local diameter of geometric graphs: we define it as the upper bound on the size of the shortest path between any pair of vertices, expressed as a function of the density of the graph around these vertices. We determine the local diameter of several well-studied graphs such as @Q-graphs, Ordered @Q-graphs and Skip List Spanners. We also show that various operations, such as path and point queries using geometric graphs as data structures, have complexities which can be expressed as local properties.

[1]  Erik D. Demaine,et al.  Proximate point searching , 2004, CCCG.

[2]  Naveed A. Sherwani,et al.  Algorithms for VLSI Physical Design Automation , 1999, Springer US.

[3]  Robert E. Tarjan,et al.  Self-adjusting binary search trees , 1985, JACM.

[4]  Joachim Gudmundsson,et al.  Ordered theta graphs , 2004, CCCG.

[5]  John Iacono,et al.  Proximate planar point location , 2003, SCG '03.

[6]  Godfried T. Toussaint,et al.  Relative neighborhood graphs and their relatives , 1992, Proc. IEEE.

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

[8]  Jan van Leeuwen,et al.  Handbook of Theoretical Computer Science, Vol. A: Algorithms and Complexity , 1994 .

[9]  Kenneth L. Clarkson,et al.  Approximation algorithms for shortest path motion planning , 1987, STOC.

[10]  R. G. Loomis,et al.  Flexible abbreviation of words in a computer language , 1963, CACM.

[11]  Jean Cardinal,et al.  Region counting graphs , 2005, EuroCG.

[12]  David Eppstein,et al.  On Nearest-Neighbor Graphs , 1992, ICALP.

[14]  Carl Gutwin,et al.  Classes of graphs which approximate the complete euclidean graph , 1992, Discret. Comput. Geom..

[15]  Kurt Mehlhorn,et al.  Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity , 1990 .

[16]  Paul Chew,et al.  There is a planar graph almost as good as the complete graph , 1986, SCG '86.

[17]  Robert E. Tarjan,et al.  Design and Analysis of a Data Structure for Representing Sorted Lists , 1978, SIAM J. Comput..

[18]  John Iacono,et al.  Proximate point location , 2003, SoCG 2003.

[19]  D. T. Lee,et al.  Generalized delaunay triangulation for planar graphs , 1986, Discret. Comput. Geom..

[20]  William Pugh,et al.  Skip lists: a probabilistic alternative to balanced trees , 1989, CACM.

[21]  Michiel H. M. Smid,et al.  Randomized and deterministic algorithms for geometric spanners of small diameter , 1994, Proceedings 35th Annual Symposium on Foundations of Computer Science.