Shortest path with a fixed number of edges

Question

Find the shortest path through a graph in efficient time, with the additional constraint that the path must contain exactly n nodes.

We have a directed, weighte

Answer 1

A rough idea of an algorithm:

Let A be the start node, and let S be a set of nodes (plus a path). The invariant is that at the end of step n, S will all nodes that are exactly n steps from A and the paths will be the shortest paths of that length. When n is 0, that set is {A (empty path)}. Given such a set at step n - 1, you get to step n by starting with an empty set S1 and

for each (node X, path P) in S
  for each edge E from X to Y in S, 
    If Y is not in S1, add (Y, P + Y) to S1
    If (Y, P1) is in S1, set the path to the shorter of P1 and P + Y

There are only n steps, and each step should take less than max(N, E), which makes the entire algorithm O(n^3) for a dense graph and O(n^2) for a sparse graph.

This algorith was taken from looking at Dijkstra's, although it is a different algorithm.