Fast constructions of lightweight spanners for general graphs

It is long known that for every weighted undirected n-vertex m-edge graph G = (V, E, ω), and every integer k ≥ 1, there exists a ((2k- 1) · (1 + ϵ))-spanner with O(n^1+1/k) edges and weight O(k · n1/k · ω(MST(G)), for an arbitrarily small constant ϵ > 0. (Here ω(MST(G)) stands for the weight of the minimum spanning tree of G.) To our knowledge, the only algorithms for constructing sparse and lightweight spanners for general graphs admit high running times. Most notable in this context is the greedy algorithm of Althöfer et al. [1993], analyzed by Chandra et al. [1992], which requires O(m· (n^1+1/k + n · log n)) time. In this article, we devise an efficient algorithm for constructing sparse and lightweight spanners. Specifically, our algorithm constructs ((2k - 1) · (1 + ϵ))-spanners with O(k · n^1+1/k) edges and weight O(k · n1/k) · ω(MST(G)), where ϵ > 0 is an arbitrarily small constant. The running time of our algorithm is O(k ·m+ min{n · log n,m· α(n)}). Moreover, by slightly increasing the running time we can reduce the other parameters. These results address an open problem by Roditty and Zwick [2004].