Floyd-Warshall Algorithm

Let the vertices of G be V = {1, 2........n} and consider a subset {1, 2........k} of vertices for some k. For any pair of vertices i, j ∈ V, considered all paths from i to j whose intermediate vertices are all drawn from {1, 2.......k}, and let p be a minimum weight path from amongst them. The Floyd-Warshall algorithm exploits a link between path p and shortest paths from i to j with all intermediate vertices in the set {1, 2.......k-1}. The link depends on whether or not k is an intermediate vertex of path p.

If k is not an intermediate vertex of path p, then all intermediate vertices of path p are in the set {1, 2........k-1}. Thus, the shortest path from vertex i to vertex j with all intermediate vertices in the set {1, 2.......k-1} is also the shortest path i to j with all intermediate vertices in the set {1, 2.......k}.

If k is an intermediate vertex of path p, then we break p down into i → k → j.

Let d_ij^(k) be the weight of the shortest path from vertex i to vertex j with all intermediate vertices in the set {1, 2.......k}.

A recursive definition is given by

FLOYD - WARSHALL (W)
 1. n ← rows [W].
 2. D0 ← W
 3. for k ← 1 to n
 4. do for i ← 1 to n     
 5. do for j ← 1 to n     
 6. do dij(k) ← min (dij(k-1),dik(k-1)+dkj(k-1) )
 7. return D(n)      

The strategy adopted by the Floyd-Warshall algorithm is Dynamic Programming. The running time of the Floyd-Warshall algorithm is determined by the triply nested for loops of lines 3-6. Each execution of line 6 takes O (1) time. The algorithm thus runs in time θ(n³ ).

Example: Apply Floyd-Warshall algorithm for constructing the shortest path. Show that matrices D^(k) and π^(k) computed by the Floyd-Warshall algorithm for the graph.

Solution:

Step (i) When k = 0

Step (ii) When k =1

Step (iii) When k = 2

Step (iv) When k = 3

Step (v) When k = 4

Step (vi) When k = 5

TRANSITIVE- CLOSURE (G)
 1. n ← |V[G]|
 2. for i ← 1 to n
 3. do for j ← 1 to n
 4. do if i = j or (i, j) ∈ E [G]
 5. the ← 1
 6. else ← 0
 7. for k ← 1 to n
 8. do for i ← 1 to n
 9. do for j ← 1 to n
 10. dod ij(k) ← 
 11. Return T(n).