Which algorithm can I use to find the next to shortest path in a graph?

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 .W

Answer 1

This is assuming you can reuse edges and nodes:

A straightfoward solution is do make an extension of the Djikstra Algorithm.

• Instead of storing for each node the smallest cost and its respective parent, store the two smallest costs (and their respective parents).

• For the priority queue, intead of storing nodes, store (node, i) pairs, so you know wether to use the 1st or 2nd path during propagation.

• Take care during the propagation phases to keep the multiple path values correctly updated.

(I might be missing some important details but the basic idea is here...)

Answer 2

When you prefer a practical solution to an academic one, here is one.

I solved this by setting a penalty to the shortest path edges and running the search again.

E.g. shortest path has length 1000, penalty is 10%, so I search for a 2nd shortest path with 1000<=length<=1100.

In the worst case I find the previous shortest path.
In the best case I find a disjunct path with the same length.
In most cases I find a path sharing some locally optimal subpaths.

Increasing the penalty forces the algorithm to find alternative routes, while decreasing makes it sharing tolerant.

When I find the 2nd shortest path, I have to subtract the sum of penalties on shared edges from the computed length to get the real length.

For k-th shortest path I set the penalty to all edges used in previous k-1 shortest paths.