Finding Min-Sum Disjoint Shortest Paths from a Single Source to All Pairs of Destinations

Abstract

Given a graph G = (V, E) with |V| = n, |E| = m, and a source node s, we consider the problem of finding two disjoint paths from s to two destination nodes t ₁ and t ₂ with minimum total length, for every pair nodes t ₁, t ₂ ∈ V–{s}. One efficient solution is to transform this problem into the problem of finding shortest pairs of disjoint paths, and use the Suurablle-Tarjan algorithm to solve the new problem in O(n ² log n) time and O(n ²) space. We present an algorithm that solves this problem in O(n ²) time and O(n ²) space, with the solution paths are implicitly represented. Given such a representation, the time necessary to explicitly construct all the solution paths is O(1) for each edge on the paths. Based on this algorithm, we present another algorithm that solves this problem in O(m log _1 + m/n )n time and O(m) space, with the compromise of longer searching time on solution paths.