A linear time algorithm for the next-to-shortest path problem on undirected graphs with nonnegative edge lengths

TL;DR

This paper presents a linear time algorithm for finding the next-to-shortest path in undirected graphs with nonnegative edge lengths, generalizing previous work to include zero-length edges.

Contribution

The paper introduces a linear time algorithm for the next-to-shortest path problem in undirected graphs with nonnegative edges, given distances from source and target.

Findings

Algorithm runs in linear time given distances from s and t.

Extends previous algorithms to include zero-length edges.

Efficiently solves the next-to-shortest path problem in undirected graphs.

Abstract

For two vertices $s$ and $t$ in a graph $G=(V,E)$, the next-to-shortest path is an $st$-path which length is minimum amongst all $st$-paths strictly longer than the shortest path length. In this paper we show that, when the graph is undirected and all edge lengths are nonnegative, the problem can be solved in linear time if the distances from $s$ and $t$ to all other vertices are given. This result generalizes the previous work (DOI 10.1007/s00453-011-9601-7) to allowing zero-length edges.