Question

I want to find the next shortest path between 2 vertices in a graph and the path has a positive cost.The next shortest path is allowed to share edges of the shortest path .Which algorithm can I use?

Answer 1

I doubt this is optimal in terms of running time but:

1. Use Dijkstra’s algorithm on graph G to get path P
2. For all edges E in path P:
3. — If G – E is not connected, continue for next edge (go to 2)
4. — Use Dijkstra’s algorithm on G – E to find path X_i
5. — If length of current X_i is shorter than all other X_i, save it
6. The X_i at the end of the for loop is the second shortest path

The second-shortest path can’t go through all edges in P, but it could go through all but one of them, potentially. I assume by “second-shortest” that you don’t use edges more than once, otherwise the second-shortest path could contain P.