We study the s-sources almost shortest paths (abbreviated 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|nρ+s.n1+ζ) for this problem that computes the paths Pu, w for all pairs (u, w) ∈ S ×V such that the length of Pu, w is at most (1 + ϵ)dG(u, w) + β(ζ, ρ, ϵ), and β(ζ, ρ, ϵ) is constant when ζ, ρ, and ϵ are arbitrarily small constants. We also devise a distributed protocol for the s-ASP problem that computes the paths Pu, w as above, and has time and communication complexities of O(s. Diam(G) + n1+ ζ/2) (respectively, O(s. Diam(G) log3 n + n1+ ζ/2 log n)) and O(|E|n ρ + s. n1+ ζ) (respectively, O(|E|nρ + s. n1+ ζ + n1+ ρ+ ζ (ρ. ζ/2)/2)) in the synchronous (respectively 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+ ζ), developed in this article. This algorithm has running time of O(|E|n ρ), which is significantly faster than the previously known algorithm given in Elkin and Peleg , whose running time is.O(n2+ ρ).We also develop the first distributed protocol for constructing (1+ϵ, β)-spanners. The communication complexity of this protocol is near optimal.
- Almost shortest paths
- Graph algorithms