I'm trying to find efficient algorithm for finding shortest path in weighted(Positive/Negative) graph with maximum edge limit.

For example, in the graph below if we demand the shortest path from A to C with restriction the path can contain at most two edges. It would be A->B->C and total cost 5 (Which is greater than A->D->E->C)

I think when the graph is (directed or undirected) has no negative edge, we can use Dijkstra with little modification, such that we will keep the number of edges in the currently discovered shortest path from source to currently used vertex for relaxation. Is my idea OK? Or there is much better something?

And how can we solve the similar problem when the graph(directed) consists of negative edges(but no negative cycle)?

I think Bellman-Ford algorithm is exactly what you need

Hi Morozov, Thank You for reply :) Can you please give me some more details about your Idea...

And if you don't mind can you please specify any pitfall to use Dijkstra when the graph does not contain any negative edge.

Calculate for each node

xand nonnegative integerk<nthe numberd_{x, k}, the length of the shortest path from the starting node toxwhich uses exactlykedges. The recurrence is similar to what you have in Dijkstra's algorithm, if edge weights are nonnegative, of course. If you have negative edges you can apply the same modification to the Bellman-Ford algorithm.Maybe there is a faster solution though?

Hi Ivan, Thank You very much :)

That's really a great and simple Idea :D