Graph Theory Questions Medium
The Floyd-Warshall algorithm is a dynamic programming algorithm used to find the shortest paths between all pairs of vertices in a weighted graph. It is particularly useful for finding the shortest path in graphs with negative edge weights.
The algorithm works by considering all possible intermediate vertices in the paths between any two vertices. It maintains a matrix, called the distance matrix, which stores the shortest distances between pairs of vertices. Initially, the distance matrix is filled with the weights of the edges in the graph.
The algorithm then iteratively updates the distance matrix by considering each vertex as a possible intermediate vertex. For each pair of vertices (u, v), it checks if the path through the intermediate vertex k yields a shorter distance than the current shortest path. If so, it updates the distance matrix accordingly.
The algorithm performs these iterations for all vertices, gradually building up the shortest paths. After the algorithm completes, the distance matrix will contain the shortest distances between all pairs of vertices.
Additionally, the algorithm can also be modified to reconstruct the actual paths between the vertices by maintaining an additional matrix, called the path matrix, which stores the intermediate vertices in the shortest paths.
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 efficient for small to medium-sized graphs, but it can become computationally expensive for large graphs.