Computing almost shortest paths

We study the s-sources almost shortest paths (shortly, s-ASP) problem. Given an unweighted graph G = (V, E), and a subset S ⊆ V of s nodes, the goal is to compute almost shortest paths between all the pairs of nodes S × V. We devise an algorithm with running time O(|E|n^ρ + s · n^1+ζ) for this problem that computes the paths P_u,w for all pairs (u, w) ε S × V such that the length of P_u,w is at most (1 + ε)d_G(u, w) + β(ζ, ρ, ε) is constant when ζ, ρ and ε are (one can choose arbitrarily small constants ζ, ρ and ε). We also devise a distributed protocol for the s-ASP problem that computes the paths P_u,w as above, and has time and communication complexities of O(s · Diam(G)+n^1+ζ/2) (resp., O(s · Diam(G) log³ n + n^1+ζ/2 log n)) and O(|E|n^ρ + s · n^1+ζ) (resp., O(|E|n^ρ + s · n^1+ζ + n^{1+ρ+ζ(ρ-ζ/2)/2})) in the synchronous (resp., asynchronous) setting. Our sequential algorithm, as well as the distributed protocol, is based on a novel algorithm for constructing (1 + ε, β(ζ, ρ, ε))-spanners of size O(n^1+δ), developed in this paper. This algorithm has running time of O(|E|n^ρ), which is significantly faster than the previously known algorithm of [20], whose running time is Õ(n^2+ρ). We also develop the first distributed protocol for constructing (1 + ε, β)-spanners. The time and communication complexities of this protocol are near-optimal.