What is the Floyd-Warshall algorithm and how does it work?

What is the Floyd-Warshall algorithm and how does it work?

The Floyd-Warshall algorithm is a dynamic programming method used to find the shortest paths in a weighted graph with positive or negative edge weights (but no negative cycles). It works by considering all pairs of nodes and systematically updating the shortest path between them.

Can you explain the steps involved in the algorithm?

Sure! The algorithm involves three nested loops. First, it initializes a distance matrix with direct distances between nodes. Then, it iteratively updates this matrix by checking if a path through an intermediate node is shorter than the current known path.

What is its time complexity?

The time complexity of the Floyd-Warshall algorithm is O(V^3), where V is the number of vertices in the graph. This makes it less efficient for very large graphs compared to other algorithms like Dijkstra's.

Are there any practical applications of this algorithm?

Yes, the Floyd-Warshall algorithm is used in network routing protocols, urban traffic planning, and in various applications where finding shortest paths between all pairs of nodes is essential.