Efficient solutions for finding vitality with respect to shortest paths

Let G = (V, E) be a connected, weighted, undirected graph such that |V| = n and |E| = m. Given a shortest path P_g(s, t) between a source node s and a sink node t in the graph G, computing the shortest path between source and sink without using a particular edge (or a particular node) in P_g(s, t) is called Replacement Shortest Path for that edge (or node). The Most Vital Edge (MVE) problem is to find an edge in P_g(s, t) whose removal results in the longest replacement shortest path. And the Most Vital Node (MVN) problem is to find a node in P_G(s, t) whose removal results in the longest replacement shortest path. In this paper for the MVE problem we describe an O(m+m´a(m´, n´)) time algorithm (α represents Inverse Ackermann function) by constructing a smaller graph L_G from G which we call Linear Graph, where n´ and m´ are the number of nodes and edges in L_G respectively. Our algorithm will also suggest a replacement shortest path for every edge in P_g(s, t) without any additional time. For the MVN problem, with integer weights, we describe an O(mα(m, n)) time algorithm. Our algorithm will also suggest a replacement shortest path for every node in P_G (s, t) without any additional time.