Good algorithm for finding the diameter of a (sparse) graph?

I have a large, connected, sparse graph in adjacency-list form. I would like to find two vertices that are as far apart as possible, that is, the diameter of the graph and two vertices achieving it.

I am interested in this problem in both the undirected and directed cases, for different applications. In the directed case, I of course care about directed distance (the shortest directed path from one vertex to another).

Is there a better approach than computing all-pairs shortest paths?

Edit: By "as far apart as possible", I of course mean the "longest shortest path" -- that is, the maximum over all pairs of vertices of the shortest distance from one to the other.

Solution 1:

Well, I've put a little bit of thought on the problem, and a bit of googling, and I'm sorry, but I can't find any algorithm that doesn't seem to be "just find all pairs shortest path".

However, if you assume that Floyd-Warshall is the only algorithm for computing such a thing (Big-Theta of |V|^3), then I have a bit of good news for you: Johnson's Algorithm for Sparse Graphs (thank you, trusty CLRS!) computes all pairs shortest paths in (Big-Oh (|V|^2 * lgV + VE)), which should be asymptotically faster for sparse graphs.

Wikipedia says it works for directed (not sure about undirected, but at least I can't think of a reason why not), here's the link.

Is there anything else about the graph that may be useful? If it can be mapped easily onto a 2D plane (so, its planar and the edge weights obey the triangle inequality [it may need to satisfy a stricter requirement, I'm not sure]) you may be able to break out some geometric algorithms (convex-hull can run in nlogn, and finding the farthest pair of points is easy from there).

Hope this helps! - Agor

Edit: I hope the link works now. If not, just google it. :)

Solution 2:

I don't know of a better method for computing diameter other than all shortest paths, but Mathematica uses the following approximation for PseudoDiameter:

• A graph geodesic is the shortest path between two vertices of a graph. The graph diameter is the longest possible length of all graph geodesics of the graph. PseudoDiameter finds an approximate graph diameter. It works by starting from a vertex u, and finds a vertex v that is farthest away from u. This process is repeated by treating v as the new starting vertex, and ends when the graph distance no longer increases. A vertex from the last level set that has the smallest degree is chosen as the final starting vertex u, and a traversal is done to see if the graph distance can be increased. This graph distance is taken to be the pseudo-diameter.

http://reference.wolfram.com/mathematica/GraphUtilities/ref/PseudoDiameter.html