A Quadratic Algorithm for Finding Next-to-Shortest Paths in Graphs

Abstract

Given an edge-weighted undirected graph G and two prescribed vertices u and v, a next-to-shortest (u, v)-path is a shortest (u, v)-path amongst all (u, v)-paths having length strictly greater than the length of a shortest (u, v)-path. In this paper, we deal with the problem of computing a next-to-shortest (u, v)-path. We propose an O(n 2 ) time algorithm for solving this problem, which significantly improves the bound of a previous one in O(n 3 ) time where n is the number of vertices in G.

Approximating the Canadian Traveller Problem with Online Randomization

“…and only if there exists a strictly ith-shortest s-t path p in G, 1 ≤ i ≤ j, such that (u, v) ∈ p; all isolated vertices in V j are eliminated. Previous studies [17,20,21] have investigated the strictly second-shortest path problem (i.e., nextto-shortest path) based on the shortest path digraph, i.e., D 1 (G) = (V 1 , A 1 ). time [17,20].…”

“…Previous studies [17,20,21] have investigated the strictly second-shortest path problem (i.e., nextto-shortest path) based on the shortest path digraph, i.e., D 1 (G) = (V 1 , A 1 ). time [17,20]. Additionally, for every vertex v = s, d 1 (s, v) and S 1 (v) can be obtained.…”

Simple paths with exact and forbidden lengths

“…If the graph is undirected, then the problem is polynomially solvable. For strictly positive edge lengths, algorithms with running times , , and were successively presented by Krasikov and Noble (), Li, Sun, and Chen (), and Kao, Chang, Wang, and Juan (). For non‐negative edge lengths, an time algorithm is presented by Zhang and Nagamochi ().…”

“…If , then the shortest simple path is a solution of the instance of the problem P ath N o (). If , then the next‐to‐shortest simple path is a solution of this instance.Corollary The following special cases of the problem P ath N o ( ) are polynomially solvable : if graph is directed and planar and arc lengths are strictly positive, then P ath N o ( ) can be solved in time (Wu & Wang, ) ; if graph is undirected and arc lengths are strictly positive, then P ath N o ( ) can be solved in time (Kao et al, ) ; if graph is undirected and arc lengths are non‐negative, then P ath N o ( ) can be solved in time (Zhang & Nagamochi, ). Let us now show that for graphs with directed cycles and non‐negative arc lengths the problem P ath N o () is difficult. Observation If graph contains directed cycles and the arc lengths are all equal to 0 but one arc length is equal to 1, then any of the problems E xact P ath (), P ath N o (), P ath ‐1‐G ap , P ath ‐2‐G aps , P ath G aps , S hort P ath G aps and L ong P ath G aps is NP‐complete in the strong sense.Proof Fortune, Hopcroft, and Wyllie () proved that the problem T wo D isjoint P aths is NP‐complete in the strong sense.…”

The next‐to‐shortest path problem on directed graphs with positive edge weights

“…For undirected graphs with positive edge weights, the first polynomial‐time algorithm was presented by Krasikov and Noble with time complexity , in which n and m are the number of vertices and edges, respectively. The time complexity has been improved several times , and the currently best result is a linear time algorithm, assuming that the distances from s and t to all other vertices are given . Hence, for undirected graphs with positive edge weights, the next‐to‐shortest path problem can be solved as efficiently as the single‐source shortest‐path problem.…”