Dijkstra's algorithm with negative weights

Question

Can we use Dijkstra\'s algorithm with negative weights?

STOP! Before you think \"lol nub you can just endlessly hop between two points and get an in

Answer 1

If you read the proof of optimality, one of the assumptions made is that all the weights are non-negative. So, no. As Bart recommends, use Bellman-Ford if there are no negative cycles in your graph.

You have to understand that a negative edge isn't just a negative number --- it implies a reduction in the cost of the path. If you add a negative edge to your path, you have reduced the cost of the path --- if you increment the weights so that this edge is now non-negative, it does not have that reducing property anymore and thus this is a different graph.

I encourage you to read the proof of optimality --- there you will see that the assumption that adding an edge to an existing path can only increase (or not affect) the cost of the path is critical.