algorithm that finds the number of the shortest paths from given node s to any other node v?

Question

i want to find an algorithm that when given a graph G=(V,E) and a node s from V, for any node v in V it finds the number of the shortest paths from s to v. i learned that BFS algorithm finds the shortest path from s to v, i just can't figure out how to use it in order to solve this problem. any help would be very appreciated.

Answer 1

Here is pseudo-code (meaning untested Python) for this.

path_count[s] = 1
length_to[s] = 0
todo = [s] # A queue
while len(todo):
    v = todo.pop(0)
    for w in edges(graph, v):
        if w not in length_to:
            length_to[w] = length_to[v] + 1
            path_count[w] = 0
            todo.append(w)

        if length_to[w] == length_to[v] + 1:
            path_count[w] += path_count[v]

After this length_to[v] gives the length of the shortest path to v and paths_to[v] gives the number of shortest paths to v . (Which is what you were looking for.)