Computing almost shortest paths

Abstract

We study the s-sources almost shortest paths (shortly, s-ASP) problem. Given an unweighted graph G = (V, E), and a subset S n V of s nodes, the goal is to compute almost shortest paths between all the pairs of nodes S x V. We devise an algorithm with running time O(¦E¦ nr + s · n1+z) for this problem that computes the paths Pu, w for all pairs (u, w) ∈ S x V such that the length of Pu, w is at most (1 + ∈) dG (u, w) + b(z, r, ∈), and b(z, r, ∈) is constant when z, r and ∈ are (one can choose arbitrarily small constants z, r and ∈).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+z/2) (resp., O(s · Diam(G) log3n + n1+z/2 log n)) and O(¦E¦ nr + s · n1+z (resp., O(¦E¦nrgr; + s · n1+z + n1+r+z(r-z/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 + ∈, b(z, r, ∈))-spanners of size O(n1+5), developed in this paper. This algorithm has running time of O(¦E¦ nr), which is significantly faster than the previously known algorithm of [20], whose running time is O(n2+r). We also develop the first distributed protocol for constructing (1 + ∈, b)-spanners. The time and communication complexities of this protocol are near-optimal.