Improved Approximation For The Directed Spanner Problem

We give an O(root n log n)-approximation algorithm for the problem of finding the sparsest spanner of a given directed graph G on n vertices. A spanner of a graph is a sparse subgraph that approximately preserves distances in the original graph. More precisely, given a graph G = (V, E) with nonnegative edge lengths d : E -> R->= 0 and a stretch k >= 1, a subgraph H = (V, E-H) is a k-spanner of G if for every edge (u, v) is an element of E, the graph H contains a path from u to v of length at most k . d(u, v). The previous best approximation ratio was (O) over tilde (n(2/3)), due to Dinitz and Krauthgamer (STOC '11).We also present an improved algorithm for the important special case of directed 3-spanners with unit edge lengths. The approximation ratio of our algorithm is (O) over tilde (n(1/3)) which almost matches the lower bound shown by Dinitz and Krauthgamer for the integrality gap of a natural linear programming relaxation. The best previously known algorithms for this problem, due to Berman, Raskhodnikova and Ruan (FSTTCS '10) and Dinitz and Krauthgamer, had approximation ratio (O) over tilde(root n).