Fast Algorithms for Constructing t-Spanners and Paths with Stretch t

The distance between two vertices in a weighted graph is the weight of a minimum-weight path between them (where the weight of a path is the sum of the weights of the edges in the path). A path has stretch t if its weight is at most t times the distance between its end points. We present algorithms that compute paths of stretch $2\leq t\leq\log n$ on undirected graphs G=(V,E) with nonnegative weights. The stretch t is of the form $t=\beta(2+\epsilon')$, where $\beta$ is integral and $\epsilon'>0$ is at least as large as some fixed $\epsilon>0$. We present an $\tilde{O}((m+k)n^{(2+\epsilon)/t})$ time randomized algorithm that finds paths between k specified pairs of vertices and an $\tilde{O}((m+ns)n^{2(1+\log_n m+\epsilon)/t})$ deterministic algorithm that finds paths from $s$ specified sources to all other vertices (for any fixed $\epsilon>0$), where n=|V| and m=|E|. This improves significantly over the slower $\tilde{O}(\min\{k,n\}m)$ exact shortest paths algorithms and a previous $\tilde{O}(mn^{64/t}+kn^{32/t})$ time algorithm by Awerbuch {et al.}\ [Proc. 34th IEEE Annual Symposium on Foundations of Computer Science, IEEE, Piscataway, NJ, 1993, pp. 638--647]. A t-spanner of a graph G is a set of weighted edges on the vertices of G such that distances in the spanner are not smaller and within a factor of t from the corresponding distances in G. Previous work was concerned with bounding the size and efficiently constructing t-spanners. We construct t-spanners of size $\tilde{O}(n^{1+(2+\epsilon)/t})$ in $\tilde{O}(mn^{(2+\epsilon)/t})$ expected time (for any fixed $\epsilon>0$), which constitutes a faster construction (by a factor of n^3+2/t /m) of sparser spanners than was previously attainable. We also provide efficient parallel constructions. Our algorithms are based on pairwise covers and a novel approach to construct them efficiently.