On Spanners and Lightweight Spanners of Geometric Graphs

We consider the problem of computing spanners of Euclidean and unit disk graphs embedded in the two-dimensional Euclidean plane. We are particularly interested in spanners that possess useful properties such as planarity, bounded degree, and/or light weight. Such spanners have been extensively studied in the area of computational geometry and have been used as the building block for constructing efficient and reliable wireless network communication topologies. We study the above problem under two computational models: the centralized and the distributed model. In the distributed model we focus on algorithms that are local. Such algorithms are suitable for the relevant applications (e.g., wireless computing). Under the centralized model, we present an $O(n\lg n)$ time algorithm that computes a bounded-degree plane spanner of a complete Euclidean graph, where $n$ is the number of points in the graph. Both upper bounds on the degree and the stretch factor significantly improve the previous bounds. We extend this algorithm to compute a bounded-degree plane lightweight spanner of a complete Euclidean graph. Under the distributed model, we give the first local algorithm for computing a spanner of a unit disk graph that is of bounded degree and plane. The upper bounds on the degree, stretch factor, and the locality of the algorithm dramatically improve the previous results, as shown in the paper. This algorithm can also be extended to compute a bounded-degree plane lightweight spanner of a unit disk graph. Our algorithms rely on structural and geometric results that we develop in this paper.

[1]  Roger Wattenhofer.  Sensor Networks: Distributed Algorithms Reloaded - or Revolutions? , 2006, SIROCCO.

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

[3]  Sriram V. Pemmaraju,et al.  Local approximation schemes for topology control , 2006, PODC '06.

[4]  Mark de Berg,et al.  Computational geometry: algorithms and applications , 1997 .

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

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

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

[8]  Michiel H. M. Smid,et al.  Diamond Triangulations Contain Spanners of Bounded Degree , 2006, ISAAC.

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

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

[11]  David Peleg,et al.  Distributed Computing: A Locality-Sensitive Approach , 1987 .

[12]  Ljubomir Perkovic,et al.  On Geometric Spanners of Euclidean and Unit Disk Graphs , 2008, STACS.

[13]  Giri Narasimhan,et al.  Approximating Geometric Bottleneck Shortest Paths , 2003, STACS.

[14]  David P. Dobkin,et al.  Delaunay graphs are almost as good as complete graphs , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[15]  Jorge Urrutia,et al.  Compass routing on geometric networks , 1999, CCCG.

[16]  Joachim Gudmundsson,et al.  Constructing Plane Spanners of Bounded Degree and Low Weight , 2005, Algorithmica.

[17]  Nathan Linial,et al.  Locality in Distributed Graph Algorithms , 1992, SIAM J. Comput..

[18]  Xiang-Yang Li,et al.  Localized Delaunay Triangulation with Application in Ad Hoc Wireless Networks , 2003, IEEE Trans. Parallel Distributed Syst..

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

[20]  Ge Xia,et al.  Computing Lightweight Spanning Subgraphs Locally , 2008 .

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

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

[23]  Gruia Calinescu.  Computing 2-Hop Neighborhoods in Ad Hoc Wireless Networks , 2003, ADHOC-NOW.

[24]  Ivan Stojmenovic,et al.  Routing with Guaranteed Delivery in Ad Hoc Wireless Networks , 1999, DIALM '99.

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

[26]  Xiang-Yang Li,et al.  Distributed construction of a planar spanner and routing for ad hoc wireless networks , 2002, Proceedings.Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies.

[27]  Xiang-Yang Li,et al.  Localized Construction of Bounded Degree and Planar Spanner for Wireless Ad Hoc Networks , 2003, DIALM-POMC '03.