Experimental study of geometric t-spanners

The construction of t-spanners of a given point set has received a lot of attention, especially from a theoretical perspective. In this article, we experimentally study the performance and quality of the most common construction algorithms for points in the Euclidean plane. We implemented the most well-known t-spanner algorithms and tested them on a number of different point sets. The experiments are discussed and compared to the theoretical results, and in several cases, we suggest modifications that are implemented and evaluated. The measures of quality that we consider are the number of edges, the weight, the maximum degree, the spanner diameter, and the number of crossings. This is the first time an extensive comparison has been made between the running times of construction algorithms of t-spanners and the quality of the generated spanners.

[1]  Giri Narasimhan,et al.  New sparseness results on graph spanners , 1992, SCG '92.

[2]  Jeffrey S. Salowe.  Constructing multidimensional spanner graphs , 1991, Int. J. Comput. Geom. Appl..

[3]  Giri Narasimhan,et al.  New sparseness results on graph spanners , 1995, Int. J. Comput. Geom. Appl..

[4]  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.

[5]  Joachim Gudmundsson,et al.  Approximate distance oracles for geometric graphs , 2002, SODA '02.

[6]  Xiang-Yang Li,et al.  Geometric Spanners for Wireless Ad Hoc Networks , 2003, IEEE Trans. Parallel Distributed Syst..

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

[8]  Gonzalo Navarro,et al.  t-Spanners as a Data Structure for Metric Space Searching , 2002, SPIRE.

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

[10]  J. Mark Keil,et al.  Approximating the Complete Euclidean Graph , 1988, Scandinavian Workshop on Algorithm Theory.

[11]  Xiang-Yang Li,et al.  Geometric spanners for wireless ad hoc networks , 2002, Proceedings 22nd International Conference on Distributed Computing Systems.

[12]  Arthur M. Farley,et al.  Spanners and message distribution in networks , 2004, Discret. Appl. Math..

[13]  Giri Narasimhan,et al.  A Fast Algorithm for Constructing Sparse Euclidean Spanners , 1997, Int. J. Comput. Geom. Appl..

[14]  Martin Zachariasen,et al.  Construction of Minimum-Weight Spanners , 2004, ESA.

[15]  Giri Narasimhan,et al.  Geometric spanner networks , 2007 .

[16]  Giri Narasimhan,et al.  Improved Algorithms for Constructing Fault-Tolerant Spanners , 2001, Algorithmica.

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

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

[19]  Robert L. Scot Drysdale,et al.  A comparison of sequential Delaunay triangulation algorithms , 1995, SCG '95.

[20]  Andrzej Lingas,et al.  Fast Approximation Schemes for Euclidean Multi-connectivity Problems , 2000, ICALP.

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

[22]  Giri Narasimhan,et al.  A new way to weigh Malnourished Euclidean graphs , 1995, SODA '95.

[23]  Gonzalo Navarro,et al.  Practical Constraction of Metric t-Spanners , 2003, ALENEX.

[24]  Joachim Gudmundsson,et al.  Fast Greedy Algorithms for Constructing Sparse Geometric Spanners , 2002, SIAM J. Comput..

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

[26]  Joachim Gudmundsson,et al.  Approximate Distance Oracles Revisited , 2002, ISAAC.

[27]  Giri Narasimhan,et al.  Optimally sparse spanners in 3-dimensional Euclidean space , 1993, SCG '93.

[28]  José Soares,et al.  Approximating Euclidean distances by small degree graphs , 1994, Discret. Comput. Geom..

[29]  Joachim Gudmundsson,et al.  On Algorithms for Computing the Diameter of a t-Spanner , 2006 .

[30]  S. Rao Kosaraju,et al.  A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields , 1995, JACM.

[31]  Andrzej Lingas,et al.  Minimum Spanning Trees , 2022 .

[32]  Michiel H. M. Smid,et al.  Computing the Greedy Spanner in Near-Quadratic Time , 2008, SWAT.

[33]  Helen C. Purchase,et al.  Which Aesthetic has the Greatest Effect on Human Understanding? , 1997, GD.

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

[35]  Pravin M. Vaidya,et al.  A sparse graph almost as good as the complete graph on points inK dimensions , 1991, Discret. Comput. Geom..

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

[37]  Kurt Mehlhorn,et al.  LEDA: a platform for combinatorial and geometric computing , 1997, CACM.

[38]  Barun Chandra.  Constructing Sparse Spanners for Most Graphs in Higher Dimensions , 1994, Inf. Process. Lett..

[39]  Robert L. Scot Drysdale,et al.  A Comparison of Sequential Delaunay Triangulation Algorithms , 1997, Comput. Geom..

[40]  Michiel H. M. Smid,et al.  Dynamic algorithms for geometric spanners of small diameter: Randomized solutions , 1999, Comput. Geom..

[41]  Joachim Gudmundsson,et al.  Improved Greedy Algorithms for Constructing Sparse Geometric Spanners , 2000, SWAT.

[42]  David P. Dobkin,et al.  On sparse spanners of weighted graphs , 1993, Discret. Comput. Geom..