A simpler and more efficient algorithm for the next-to-shortest path problem

TL;DR

This paper introduces a new linear-time algorithm for the next-to-shortest path problem in undirected graphs, significantly improving efficiency over previous methods, especially for specific graph types.

Contribution

The paper presents a novel linear-time algorithm for the next-to-shortest path problem, improving upon prior quadratic-time solutions for sparse graphs.

Findings

Algorithm runs in O(|V| log |V| + |E|) time for general graphs.

Linear time complexity achieved for unweighted, planar, and integer edge length graphs.

Significant efficiency improvement over previous algorithms.

Abstract

Given an undirected graph $G=(V,E)$ with positive edge lengths and two vertices $s$ and $t$, the next-to-shortest path problem is to find an $st$-path which length is minimum amongst all $st$-paths strictly longer than the shortest path length. In this paper we show that the problem can be solved in linear time if the distances from $s$ and $t$ to all other vertices are given. Particularly our new algorithm runs in $O(|V|\log |V|+|E|)$ time for general graphs, which improves the previous result of $O(|V|^2)$ time for sparse graphs, and takes only linear time for unweighted graphs, planar graphs, and graphs with positive integer edge lengths.