All-Pairs Shortest Path Algorithm

There is a simple dynamic programming method for finding the shortest path between any two nodes. It works as follows:

  1. Start out with the graph in the form of an adjacency matrix.
  2. For each k:
  3.     for each pair i,j where i,j are not k.
  4.         compare the length of the edge from i to j with the sum of the length of the edge from i to k and the edge from k to j.
  5.          if the sum of the i to k and k to j edge is shorter, set the "shortest" path from i to j to this sum and remember k
  6.     end for
  7.  If we have repeated this for every k we will have the shortest path in each spot and we will know how we got there.

Example:

Edge/Edge
 0
 1
 2
 3
 4
0
 0
 1
 1000
 2
 4
1
 1
 0
 78
 6
 3
2
 1000
 78
 0
 2
 4
3
 2
 6
 2
 0
 1
4
 4
 3
 4
 1
 0
After doing the algorithm above, we get:

Edge/Edge
 0
 1
 2
 3
 4
0
 0
 1
 4,3
 2
 3,3
1
 1
 0
 5,3
 3,0
 3
2
 4,3
 5,3
 0
 2
 3,3
3
 2
 3,0
 2
 0
 1
4
 3,3
 3
 3,3
 1
 0