Question

Suppose G is a DAG,i.e., a directed acyclic graph. Lets and t be two nodes in the graph. Describe an O(n+m) algorithm that computes the number of paths from s to t . (Hint: start by topologically ordering G.)

Answer 1

For example, in the illustration below, each stage of the DAG increases the total number of paths by a multiple of 3. This exponential growth can be handled with dynamic programming.

A simple modification of DFS will compute this given as

def dfs(u, t):
    if u == t:
        return 1
    else:
        if not u.npaths:
            # assume sum returns 0 if u has no children
            u.npaths = sum(dfs(c, t) for c in u.children)
        return u.npaths

It is not difficult to see that each edge is looked at only once, hence a runtime of O(V+E).