Faster replacement paths algorithms in case of edge or node failure for undirected, positive integer weighted graphs

In this paper, we study the following replacement paths problem for undirected graphs in case of edge or node failures.1.In the edge failure problem, for each edge e   on a shortest s–ts–t path in G  , we are required to find a shortest s–ts–t path in G−eG−e.2.In the node failure problem, for each node v   on a shortest s–ts–t path, we need to report a shortest s–ts–t path in G−vG−v.3.In the detour critical problem, for each edge (u,v)(u,v) on a shortest s–ts–t path, we have to report a shortest u–tu–t path in G−(u,v)G−(u,v). If m is the number of edges and d is the distance between s and t  , which in turn will be bounded by the diameter of the graph, the proposed algorithm for all these problems takes O(m+d2)O(m+d2) time, for graphs with integer weights (on the RAM model) and on planar graphs. For typically dense graphs, or graphs with small diameter (formally, when d=O(m)), the algorithms take linear time.