A Simple Algorithm For Replacement Paths Problem

TL;DR

This paper introduces a simple, efficient algorithm for the replacement paths problem in weighted graphs, capable of computing shortest alternative paths after edge or node failures with improved time complexity.

Contribution

The paper presents a novel, easy-to-implement algorithm that solves both edge and node replacement path problems in optimal time relative to shortest path computations.

Findings

Algorithm runs in O(T_{SPT}(G)+m+l^2) time

Uses O(m+l^2) space

Effective for large graphs with positive weights

Abstract

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+l^2) time and O(m+l^2) 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.