I just thought about something.
If I want to find shortest path between A and B with negative edges, but no negative cycle.
How about we start Dijkstra with source A and then with source B and return the minimum of the two results.
I was thinking and I cannot find a counter-example.
Any ideas?
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counter-example:
1 2 5
1 3 2
2 3 -7
3 4 -5
3 5 1
4 5 2
Path from 1 to 5.
3-4-5 <= negative cycle
There is no 5-3 edge, edges are oriented. If not — so any negative edge is a cycle itselves.
Yes, I made a mistake. The weight of edge 3 4 should be -3.
OK, agreed :) Thanks!
Of course, your idea is wrong:
Path from A to B. (ans is 190 in both cases)
Search "dijkstra with potentials" or "ford-bellman".
But only if you are going to use Dijkstra's algorithm for finding min cost max flow.
In general, it doesn't work?
I can't think of any other use for it. To use Dijkstra with potentials, you need to use Ford-Bellman beforehand, to set initial potentials. To find shortest paths just once, you can just use Ford-Bellman. Potentials are needed to run Dijkstra (instead of Ford-Bellman) repetitively on changing graph.
Understood. Thanks!