On sparse spanners of weighted graphs

Given a graphG, a subgraphG' is at-spanner ofG if, for everyu,v ɛV, the distance fromu tov inG' is at mostt times longer than the distance inG. In this paper we give a simple algorithm for constructing sparse spanners for arbitrary weighted graphs. We then apply this algorithm to obtain specific results for planar graphs and Euclidean graphs. We discuss the optimality of our results and present several nearly matching lower bounds.

[1]  J. M. S. S. Pereira,et al.  A note on the tree realizability of a distance matrix , 1969 .

[2]  J. Q. Longyear.  Regular d-valent graphs of girth 6 and 2(d2−d+1) vertices , 1970 .

[3]  B. Bollobás,et al.  Extremal Graph Theory , 2013 .

[4]  Bela Bollobas,et al.  Graph theory , 1979 .

[5]  Robert E. Tarjan,et al.  Data structures and network algorithms , 1983, CBMS-NSF regional conference series in applied mathematics.

[6]  A. Dress.  Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: A note on combinatorial properties of metric spaces , 1984 .

[7]  Baruch Awerbuch,et al.  Complexity of network synchronization , 1985, JACM.

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

[9]  A. Dress,et al.  Reconstructing the shape of a tree from observed dissimilarity data , 1986 .

[10]  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).

[11]  Ingo Althöfer,et al.  On optimal realizations of finite metric spaces by graphs , 1988, Discret. Comput. Geom..

[12]  Eli Upfal,et al.  A tradeoff between space and efficiency for routing tables , 1988, STOC '88.

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

[14]  Carl Gutwin,et al.  The Delauney Triangulation Closely Approximates the Complete Euclidean Graph , 1989, WADS.

[15]  Baruch Awerbuch,et al.  Compact distributed data structures for adaptive routing , 1989, STOC '89.

[16]  David Peleg,et al.  An Optimal Synchronizer for the Hypercube , 1989, SIAM J. Comput..

[17]  Gautam Das,et al.  WHICH TRIANGULATIONS APPROXIMATE THE COMPLETE GRAPH? , 2022 .

[18]  Andrzej Lingas,et al.  Ther Are Planar Graphs Almost as Good as the Complete Graphs and as Short as Minimum Spanning Trees , 1989, Optimal Algorithms.

[19]  Jose Augusto Ramos Soares,et al.  Graph Spanners: a Survey , 1992 .

[20]  Baruch Awerbuch,et al.  Routing with Polynomial Communication-Space Trade-Off , 1992, SIAM J. Discret. Math..