A Simple Algorithm For Replacement Paths Problem

Let G=(V,E)(|V|=n and |E|=m) be an undirected graph with positive edge weights. Let P{G}(s, t) be a shortest s-t path in G. Let l be the number of edges in P{G}(s, t). The \emph{Edge Replacement Path} problem is to compute a shortest s-t path in G{e}, for every edge e in P{G}(s, t). The \emph{Node Replacement Path} problem is to compute a shortest s-t path in G{v}, for every vertex v in P{G}(s, t). In this paper we present an O(T{SPT}(G)+m+l²⁾ time and O(m+l²⁾ space algorithm for both the problems. Where, T{SPT}(G) is the asymptotic time to compute a single source shortest path tree in G. The proposed algorithm is simple and easy to implement.