How can you use the Floyd-Warshall algorithm to find shortest paths in a graph?

The Floyd-Warshall algorithm is a powerful tool for finding shortest paths in graphs, accommodating various edge weights. Initialize a matrix with direct distances, iteratively consider vertices as intermediates, and update distances if shorter paths are found. The resulting matrix reveals the shortest paths between all pairs of vertices. Detect negative cycles by checking diagonal elements. While versatile, its O(V^3) time complexity may impact efficiency in large graphs compared to other algorithms.

Complexity : N^3 Space Complexity : Node*Node Ideal use case: When distance from each pair node is given or can be calculated Concept(brief): Go from each node to another via every possible nodes and store the shortest one. Do it for every node and you got it

def floyd_warshall(graph): num_nodes = len(graph) # Initialize distance matrix with the given graph distance_matrix = [row[:] for row in graph] # Main algorithm loop for k in range(num_nodes): for i in range(num_nodes): for j in range(num_nodes): # Update the distance matrix if a shorter path is found distance_matrix[i][j] = min(distance_matrix[i][j], distance_matrix[i][k] + distance_matrix[k][j]) return distance_matrix

The algorithm does not work on negative cycle. Has more space complexity and run time complexity than other shortest path finding. If we don't need node to node distance than using it is a lose

Start by initializing a distance matrix that reflects the initial distances based on the graph's edges. Next, iterate through all vertices to update the matrix using the formula distance[i][j] = min(distance[i][j], distance[i][k] + distance[k][j]), with 'k' as the intermediate vertex. Repeat this process until all vertices have been considered as intermediates. The final distance matrix will then hold the shortest paths between all pairs of vertices. If needed, path reconstruction can be performed by keeping track of the intermediate vertices used during updates. This algorithm is suitable for both directed and undirected graphs, including those with negative edge weights, making it efficient for dense graphs.