Generating sparse spanners for weighted graphs

Abstract

we give a very 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. I N T R O D U C T I O NLet G = (V, E) be a connected n-vertex graph with arbitrary positive edge weights.A subgraph G I --(V, E ~) is a t-spanner if, between any pair of vertices the distance in G ~ is at most t times longer than the distance in G. The vMue of t is the stretch factor a~ssociated with Gq We consider the problem of determining t-spanners for graphs where the spanners are sparse and t is a constant independent of the size of the graph.Sparsity will be measured according to two criteria. Let Weight(G) denote the sum of all edge weights of graph G, and Size(G) denote the number of edges. A graph is sparse in size if it has few edges. Similarly, a graph is sparse in weight if its total edge weight is small. Our results separate graphs into classes where spanners with linearly many edges achieve constant stretch factors, and classes where a non-linear number of edges are necessary.Problems of this type appear in numerous applications. Spanners appear to be the underlying graph structure in various constructions in distributed systems and communication networks [Aw~ PU1, PU]. They also appear in biology in the process of reconstructing phylogenetic trees from matrices, whose entries represent genetic distances among contemporary living species [BD]. Robotics researchers have studied spanners under the constraints of Euclidean geometry, where vertices of the graph are points in space, and edges are line segments joining pairs of points [C, DFS, D J, K, KG, LL].

t-Spanners for metric space searching

“…It was shown [1,2] that these techniques produce t-spanners with n 1+O( 1 t−1 ) edges on general graphs of n nodes.…”

“…As it regards to Euclidean t-spanners, that is, the subclass of metric t-spanners where objects are points in a D-dimensional space with Euclidean distance, much better results exist [21,1,2,26,25,36], showing that one can build tspanners with O(n) edges in O(n log D−1 n) time. These results, unfortunately, make heavy use of coordinate information and cannot be extended to general metric spaces.…”

“…Then, in the sub-tspanner represented by p 2 (stsp 2 ), we choose the object u closest to p 1 , and insert it into the sub-t-spanner represented by p 1 (stsp 1 ) verifying that all the distances towards V 1 are t-estimated. Note that this is equivalent to using the incremental algorithm, where we insert u into the growing t-spanner stsp 1 . We continue with the second closest and repeat the procedure until all the nodes in stsp 2 are inserted into stsp 1 .…”

“…Table 2 shows the least squares fittings on the data using the model |E| = a · e cD/t α · n 1+ b t−1 and time = a · e cD/t β · n 2+ b t−1 microseconds. We chose the edge model according to the analytical results of [1,2], with a slight correction to take into account the effect of the dimensionality. As it can be seen, tspanner sizes are slightly super-linear for all our construction algorithms, and times are slightly super-quadratic for t-Spanner 2, 3 and 4.…”

“…The analytical results of [1,2] show that the size of a t-spanner built over a general graph of n nodes is n 1+O( 1 t−1 ) . The terms e cD/t α and e cD/t β represent the usual exponential dependence on the dimension of the metric space (which is also usually hidden in the constant of the big-O notation).…”