Floyd-Warshall Algorithm

External

• https://www.coursera.org/learn/algorithms-npcomplete/lecture/VQStd/problem-definition
• https://www.coursera.org/learn/algorithms-npcomplete/lecture/3BBkw/optimal-substructure
• https://www.coursera.org/learn/algorithms-npcomplete/lecture/WKb60/the-floyd-warshall-algorithm

Internal

• All-Pairs Shortest Path Algorithms
• Dynamic Programming
• Bellman-Ford Shortest-Path Algorithm
• Johnson's Algorithm

Overview

The Floyd-Warshall algorithm computes all-pairs shortest path in a directed graph with arbitrary edge length, include negative lengths. The only case that does not make sense is whether the graph has negative cost cycles. In this case there is no shortest path, because the cycle can be followed an infinite number of times, leading to negative infinity length paths. If this is the case, the Floyd-Warshall will detect the situation in its final result, by scanning the diagonal of the matrix it computes.

The running time is O(n³).

Algorithm

We assume that the vertices are labeled from 1 to n.

Let A = 3D array, indexed by i, j, k
# Initialize the tridimensional matrix with the trivial cases
 
                            │ 0 if i == j
for all i,j ∈ V, A[i,j,0] = │ cij if i≠j and (i, j) ∈ E
                            │ +∞ if i≠j and (i,j) ∉ E

for k = 1 to n:
  for i = 1 to n:
    for j = 1 to n:
      A[i,j,k] = max(A[i,j,k-1], A[i,k,k-1] + A[k,j,k-1])
 

If the graph has a negative cost cycle, the diagonal of the matrix A[i,i,n] < 0 ofr at least one i ∈ V, at the end of the algorithm.