Path between two nodes

igraph, another graph module for Python can calculate all the shortest paths between a given pair of nodes. Calculating all the paths does not make sense as you have infinitely many such paths.

An example for calculating all the shortest paths from vertex 0:

>>> from igraph import Graph
>>> g = Graph.Lattice([10, 10], circular=False)
>>> g.get_all_shortest_paths(0)
[...a list of 3669 shortest paths starting from vertex 0...]

If you have igraph 0.6 or later (this is the development version at the time of writing), you can restrict the result of get_all_shortest_paths to a given end vertex as well:

>>> g.get_all_shortest_paths(0, 15)
[[0, 1, 2, 3, 4, 14, 15],
 [0, 1, 2, 12, 13, 14, 15],
 [0, 10, 11, 12, 13, 14, 15],
 [0, 1, 11, 12, 13, 14, 15],
 [0, 1, 2, 3, 13, 14, 15],
 [0, 1, 2, 3, 4, 5, 15]]

Of course you have to be careful; for instance, assume that you have a 100 x 100 grid graph (that can easily be generated by Graph.Lattice([100, 100], circular=False) in igraph). The number of shortest paths leading from the top left node to the bottom right node equals the number of possibilities to choose 100 elements out of 200 (proof: the length of the shortest path there has 200 edges, 100 of which will go “horizontally” in the grid and 100 of which will go “vertically”). This probably does not fit into your memory, therefore even calculating all the shortest paths between these two nodes is not really feasible here.

If you really need all the paths between two nodes, you can rewrite the function given on the webpage you mentioned using igraph, which will probably be faster than a pure Python solution as igraph’s core is implemented in C:

def find_all_paths(graph, start, end, path=[]):
    path = path + [start]
    if start == end:
        return [path]
    paths = []
    for node in set(graph.neighbors(start)) - set(path):
        paths.extend(find_all_paths(graph, node, end, path))
    return paths

It can be optimized more by converting the graph to an adjacency list representation first as it would spare repeated calls to graph.neighbors:

def find_all_paths(graph, start, end):
    def find_all_paths_aux(adjlist, start, end, path):
        path = path + [start]
        if start == end:
            return [path]
        paths = []
        for node in adjlist[start] - set(path):
            paths.extend(find_all_paths_aux(adjlist, node, end, path))
        return paths

    adjlist = [set(graph.neighbors(node)) for node in xrange(graph.vcount())]
    return find_all_paths_aux(adjlist, start, end, [])

Edit: fixed first example to work in igraph 0.5.3 as well, not only in igraph 0.6.