Efficient Solutions For Finding Vitality With Respect To Shortest Paths

Let G = (V; E) be a connected, weighted, undirected graph such that vertical bar V vertical bar = n and vertical bar E vertical bar = 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' alpha (m'; n')) time algorithm (alpha 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 (m alpha(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.