Question

I have a directed weighted graph G, with V vertices and E edges. Given two nodes in the graph, let’s say A, and B, and given the weight of an edge A-B denoted as w(A, B), I need to find a node C so that max(w(A, C), w(B, C)) is minimal among all possibilities. By possibilities I mean all the values C can take. I don’t know if it is completely clear, if it’s not, I’ll try to be more precise.

Answer 1

If by w(A, C) you really mean just the weight of an edge, then check all the nodes directly connected to A in turn, for total cost at worst the size of the graph, which is about as good as you could expect, assuming that you always need to read in the graph.

If by w(A, C) you mean the cost of the least cost path from A to C note that most path-finding algorithms, like http://en.wikipedia.org/wiki/Dijkstra%27s_algorithm, in fact calculate the cost of the least cost path from A to every other node. You can solve your problem by looking at each node in turn if you have both the costs of getting from A to each node and the costs of getting from each node to B.

So do one run to work out the costs from A to every other node, then reverse the edges in the node and do another run to work out the least cost path from B to every other node in the reversed graph. Then for each node you have the cost of the least cost w(A, C) and the least cost w(C, B) so you can check each node in turn to see which is the best.

If your graph contains cycles, then you need something like http://en.wikipedia.org/wiki/Floyd%E2%80%93Warshall_algorithm. If it has negative cycles, you are going to have a problem.