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

There is actually an algorithm which uses Dijkstra's algorithm in a negative path environment; it does so by removing all the negative edges and rebalancing the graph first. This algorithm is called 'Johnson's Algorithm'.

The way it works is by adding a new node (lets say Q) which has 0 cost to traverse to every other node in the graph. It then runs Bellman-Ford on the graph from point Q, getting a cost for each node with respect to Q which we will call q[x], which will either be 0 or a negative number (as it used one of the negative paths).

E.g. a -> -3 -> b, therefore if we add a node Q which has 0 cost to all of these nodes, then q[a] = 0, q[b] = -3.

We then rebalance out the edges using the formula: weight + q[source] - q[destination], so the new weight of a->b is -3 + 0 - (-3) = 0. We do this for all other edges in the graph, then remove Q and its outgoing edges and voila! We now have a rebalanced graph with no negative edges to which we can run dijkstra's on!

The running time is O(nm) [bellman-ford] + n x O(m log n) [n Dijkstra's] + O(n^2) [weight computation] = O (nm log n) time

More info: http://joonki-jeong.blogspot.co.uk/2013/01/johnsons-algorithm.html