Partitioned neighborhood spanners of minimal outdegree

A geometric spanner with vertex set P IRD is a sparse approximation of the complete Euclidean graph determined by P. We introduce the notion of partitioned neighborhood graphs (PNGs), unifying and generalizing most constructions of spanners treated in literature. Two important parameters characterizing their properties are the outdegree k IN and the stretch factor f 1 describing the ‘quality’ of approximation. PNGs have been thuroughly investigated with respect to small values of f . We, on the other hand, present in this work results about feasable values of k — so to say the other extreme. The aim of minimizing this parameter rather than the first one arises from two observations: a) It determines the amount of space required for storing PNGs. b) Many algorithms employing a (previously constructed) spanner have running times depending on its outdegree. Our results include, for fixed dimensions D as well as asymptotically, upper and lower bounds on this optimal value of k. The upper bounds are constructive and yield efficient algorithms for actually computing the corresponding graphs. 1Partially supported by EU ESPRIT Long Term Research Project 20244 (ALCOM-IT) and DFG Grants Me872/7-1 and Me872/4-1

[1]  Tamás Lukovszki,et al.  Geometric Searching in Walkthrough Animations with Weak Spanners in Real Time , 1998, ESA.

[2]  L. Sandgren.  On convex cones , 1954 .

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

[4]  Friedhelm Meyer auf der Heide,et al.  Dynamic Data Structures for Realtime Management of Large Geormetric Scences (Extended Abstract) , 1997, ESA.

[5]  Michiel Smid,et al.  Closest-Point Problems in Computational Geometry , 2000, Handbook of Computational Geometry.

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

[7]  David Eppstein,et al.  Spanning Trees and Spanners , 2000, Handbook of Computational Geometry.

[8]  Kasturi R. Varadarajan.  A divide-and-conquer algorithm for min-cost perfect matching in the plane , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[9]  Tamás Lukovszki,et al.  A Network Based Approach for Realtime Walkthrough of Massive Models , 1998, Algorithm Engineering.

[10]  Jeffrey S. Salowe.  On Euclidean spanner graphs with small degree , 1992, SCG '92.

[11]  Michael Ian Shamos,et al.  Computational geometry: an introduction , 1985 .

[12]  Giuseppe Liotta,et al.  A robust region approach to the computation of geometric graphs , 1998 .

[13]  Satish Rao,et al.  Approximating geometrical graphs via “spanners” and “banyans” , 1998, STOC '98.

[14]  Michiel H. M. Smid,et al.  Lower bounds for computing geometric spanners and approximate shortest paths , 1996, Discret. Appl. Math..

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

[16]  Gautam Das,et al.  Constructing Degree-3 Spanners with Other Sparseness Properties , 1993, ISAAC.

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

[18]  R. V. Benson.  Euclidean Geometry and Convexity , 1966 .

[19]  MatoušekJiří.  Geometric range searching , 1994 .

[20]  Michel Pocchiola,et al.  Advances in Discrete and Computational Geometry , 1998 .

[21]  Michiel H. M. Smid,et al.  Euclidean spanners: short, thin, and lanky , 1995, STOC '95.

[22]  E. Szpilrajn.  Sur l'extension de l'ordre partiel , 1930 .

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

[24]  C. A. Rogers.  Covering a sphere with spheres , 1963 .

[25]  David P. Dobkin,et al.  Delaunay graphs are almost as good as complete graphs , 1990, Discret. Comput. Geom..