Fast Constructions of Light-Weight Spanners for General Graphs
  To our knowledge, there are only two known algorithms for constructing sparse and light spanners for general graphs. One of them is the greedy algorithm of Alth$\ddot{o}$fer et al. \cite{ADDJS93}, analyzed by Chandra et al. in SoCG'92. The greedy algorithm consructs, for every \emph{weighted} undirected $n$-vertex $m$-edge graph $G = (V,E)$ and any integer $k \ge 1$, a $(2k-1)$-spanner with $O(n^{1 + 1/k})$ edges and weight $O(k \cdot n^{(1+\eps)/k}) \cdot \omega(MST(G))$, for any $\eps > 0$. The drawback of the greedy algorithm is that it requires $O(m \cdot (n^{1 + 1/k} + n \cdot \log n))$ time. The other algorithm is due to Awerbuch et al. \cite{ABP91}. It constructs $O(k)$-spanners with $O(k \cdot n^{1 + 1/k} \cdot \Lambda)$ edges, weight $O(k^2 \cdot n^{1/k} \cdot \Lambda) \cdot \omega(MST(G))$, within time $O(m \cdot k \cdot n^{1/k} \cdot \Lambda)$, where $\Lambda$ is the logarithm of the aspect ratio of the graph. The running time of both these algorithms is unsatisfactory. Moreover, the usually faster algorithm of \cite{ABP91} pays for the speedup by significantly increasing both the stretch, the sparsity, and the weight of the resulting spanner.   In this paper we devise an efficient algorithm for constructing sparse and light spanners. Specifically, our algorithm constructs $((2k-1) \cdot (1+\eps))$-spanners with $O(k \cdot n^{1 + 1/k})$ edges and weight $O(k \cdot n^{1/k}) \cdot \omega(MST(G))$, where $\eps > 0$ is an arbitrarily small constant. The running time of our algorithm is $O(k \cdot m + \min\{n \cdot \log n,m \cdot \alpha(n)\})$. Moreover, by slightly increasing the running time we can reduce the other parameters. These results address an open problem from the ESA'04 paper by Roditty and Zwick \cite{RZ04}. 
Michael Elkin and Shay Solomon
cs.DS