Floyd-Warshall Algorithm in Python

The Floyd-Warshall algorithm, named after its creators Robert Floyd and Stephen Warshall, is fundamental in computer science and graph theory. It is used to find the shortest paths between all pairs of nodes in a weighted graph. This algorithm is highly efficient and can handle graphs with both positive and negative edge weights, making it a versatile tool for solving a wide range of network and connectivity problems.

The Floyd Warshall Algorithm is an all-pair shortest path algorithm, unlike Dijkstra and Bellman-Ford which are single source shortest path algorithms. This algorithm works for both the directed and undirected weighted graphs. However, it does not work for the graphs with negative cycles (where the sum of the edges in a cycle is negative). It follows the Dynamic Programming approach to check every possible path going via every possible node to calculate the shortest distance between every pair of nodes.

Algorithm

• Initialize the solution matrix the same as the input graph matrix as a first step. 
• Then update the solution matrix by considering all vertices as an intermediate vertex. 
• The idea is to pick all vertices one by one and update all shortest paths which include the picked vertex as an intermediate vertex in the shortest path. 
• When we pick vertex number k as an intermediate vertex, we already have considered vertices {0, 1, 2, .. k-1} as intermediate vertices. 
• For every pair (i, j) of the source and destination vertices respectively, there are two possible cases. 
  • k is not an intermediate vertex in the shortest path from i to j. We keep the value of dist[i][j] as it is. 
  • k is an intermediate vertex in the shortest path from i to j. We update the value of dist[i][j] as dist[i][k] + dist[k][j], if dist[i][j] > dist[i][k] + dist[k][j]

Illustration:

# Solves the all-pairs shortest path
# problem using Floyd Warshall algorithm
def floydWarshall(graph):
    V = len(graph)

    # Add all vertices one by one to
    # the set of intermediate vertices.
    for k in range(V):

        # Pick all vertices as source one by one
        for i in range(V):

            # Pick all vertices as destination
            # for the above picked source
            for j in range(V):

                # If vertex k is on the shortest path from
                # i to j, then update the value of graph[i][j]

                if ((graph[i][j] == -1 or 
                    graph[i][j] > (graph[i][k] + graph[k][j]))
                    and (graph[k][j] != -1 and graph[i][k] != -1)):
                    graph[i][j] = graph[i][k] + graph[k][j]

if __name__ == "__main__":
    graph = [
        [0, 4, -1, 5, -1],
        [-1, 0, 1, -1, 6],
        [2, -1, 0, 3, -1],
        [-1, -1, 1, 0, 2],
        [1, -1, -1, 4, 0]
    ]
    
    floydWarshall(graph)
    for i in range(len(graph)):
        for j in range(len(graph)):
            print(graph[i][j], end=" ")
        print()

0 4 5 5 7 
3 0 1 4 6 
2 6 0 3 5 
3 7 1 0 2 
1 5 5 4 0 

Time Complexity: O(V³), where V is the number of vertices in the graph and we run three nested loops each of size V

Auxiliary Space: O(V²), to create a 2-D matrix in order to store the shortest distance for each pair of nodes.

Read here for detailed analysis: complexity analysis of the Floyd Warshall algorithm