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.
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].
We extend previous algorithmic research to a class of regular tandem repeats (RegTRs). We formally define RegTRs, as well as two important subclasses: VLTRs and MPTRs. We present algorithms for identification of TRs in these classes. Furthermore, our algorithms identify degenerate VLTRs and MPTRs: repeats containing substitutions, insertions and deletions. To illustrate our work, we present results of our analysis for two difficult regions in cattle and human data which reflect practical occurrences of these subclasses in GenBank sequence data. In addition, we show the applicability of our algorithmic techniques for identifying Alu sequences, gene clusters and other distant regions of similarity. We illustrate this with an example from yeast chromosome I.
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