What if the only negative edge costs are coming from the initial node? Will the algorithm still work?
I feel like yes, because I can't think of a counter-example, but I'm having trouble proving it. Is there a counter-example?
Negative edges are a problem for Dijkstra's because there's no guarantee that the edge you pick produces the shortest path if there is an edge you can pick later that is largely negatively weighted. But if the only negative edges are coming out of the initial node, I don't see the problem.
I'm not looking for an algorithm. I'm looking for some insight into the Dijkstra's.
I'm talking about a directed graph, if that makes a difference.