8

I'm using Bellman-Ford to find the shortest path through a graph with some negative weights. The graph has no possibility of loops and no bi-directional connections. I'd like to find the K shortest paths through the graph, where the paths share no nodes in common. Is there an algorithm I can look up to learn how to do this? Simple implementation is more important than speed at the moment.

Added: Thanks for comments. To be clear, I'm looking for the top K ways to get from a specified start node to a specified end node, with no other nodes in common. I need a global optimum; sequentially finding the best and deleting nodes does not give a satisfactory result. This one: https://en.wikipedia.org/wiki/Yen%27s_algorithm, gives the flavor of what I'm talking about, but in this case it requires non-negative edge costs and it also allows nodes to be shared.

Mastiff
  • 2,101
  • 2
  • 23
  • 38
  • 1
    I suppose the graph can be assumed to be connected though? – Codor Oct 22 '15 at 18:32
  • 1
    K shortest paths that share no nodes in common, as in the K shortest paths that connect two vertices and share only those two vertices? If the graph is loopless, could you exhaust all paths and take the shortest K? – opticaliqlusion Oct 22 '15 at 18:33
  • 2
    So you have a directed acyclic graph? Is what you're doing now to repeatedly find a shortest path and delete the interior nodes, or are you interested in a global optimization? – David Eisenstat Oct 22 '15 at 18:58
  • Bellman-Ford keeps the shortest distance and a corresponding back-pointer for each vertex in the graph. Perhaps you can just keep the top `k` shortest distances and corresponding back-pointers instead, but otherwise run the algorithm unchanged. – Daniel Wagner Oct 22 '15 at 18:59
  • 1
    When you say the top K ways, you mean that the sum is minimized or that the first is the minimum, then the second... In other words, for K = 2, is it better to have paths with lenghts -5, -1 or -4, -3? – AlexAlvarez Oct 22 '15 at 20:00
  • 1
    The goal would be to minimize the sum of costs. Thanks. – Mastiff Oct 22 '15 at 21:31
  • How big the K is? How big the graph is? Is it [sparse, dense](http://stackoverflow.com/questions/12599143/what-is-the-distinction-between-sparse-and-dense-graphs)? – Андрей Беньковский Oct 22 '15 at 22:07

1 Answers1

3

I think that the problem can be solved finding a Minimum Cost Flow.

Let's transform the graph in the following way:

  1. Replace each node v other than source and sink with two nodes v1 and v2 connected by an edge of weight 0 from v1 to v2. The incoming edges of the former v enter to v1 and the outgoing leave from v2. With this the problem is equivalent to not using those edges more than once.

  2. Set capacity 1 to all the edges.

Finding a flow of value K will give you K paths that don't share a node (because of putting the capacity to 1 in those new edges). So if this flow is a minimum cost flow, you will have that those K paths also have the minimum possible sum of costs.

This is assuming that you don't have an edge connecting the source and the sink directly. Check for that corner case separately.

AlexAlvarez
  • 811
  • 5
  • 11
  • Thanks, do you have a recommended algorithm for solving the min cost flow problem? – Mastiff Oct 23 '15 at 03:16
  • I would suggest to apply the Shortest Augmenting Path algorithm for being quite easy to code, but using Bellman-Ford instead of Dijkstra because your graph contains negative edges. – AlexAlvarez Oct 23 '15 at 17:43