Shortest Path For A Dag

Question

I have a graph with an s and t vertex that I need to find the shortest path between. The graph has a lot of special properties that I would like to capitalize on:

Answer 1

I'm going to go against my intuition and assume this isn't homework. You have to take advantage of the information that a topological ordering gives you. Whenever you examine the node n in a topological ordering, you have the guarantee that you've already traversed every possible path to n. Using this it's clear to see that you can generate the shortest path with one linear scan of a topological ordering (pseudocode):

Graph g
Source s
top_sorted_list = top_sort(g)

cost = {} // A mapping between a node, the cost of its shortest path, and 
          //its parent in the shortest path

for each vertex v in top_sorted_list:
  cost[vertex].cost = inf
  cost[vertex].parent = None

cost[s] = 0

for each vertex v in top_sorted_list:
   for each edge e in adjacensies of v:
      if cost[e.dest].cost > cost[v].cost + e.weight:
        cost[e.dest].cost =  cost[v].cost + e.weight
        e.dest.parent = v

Now you can look up any shortest path from s to a destination. You'd just need to look up the destination in the cost mapping, get it's parent, and repeat this process until you get a node whose parent is s, then you have the shortest path.