What is the primary use of the Bellman-Ford algorithm?

The Bellman-Ford algorithm is primarily used to find the shortest paths from a single source vertex to all other vertices in a weighted graph. It is particularly useful in graphs where the edges can have negative weights. Unlike Dijkstra's algorithm, which also finds shortest paths, Bellman-Ford can handle graphs with negative weight edges, as long as there are no negative weight cycles that are reachable from the source vertex.

By repeatedly relaxing the edges of the graph, the algorithm iteratively updates the shortest path estimates to each vertex. It performs this operation for a number of iterations equal to the total number of vertices minus one, which ensures that all possible shortest paths are explored. If, after these iterations, there is still a way to reduce the path length, it suggests the presence of a negative weight cycle.

This capability makes the Bellman-Ford algorithm essential in many applications, such as network routing protocols, where the costs or weights may vary dynamically.

The other options do not align with the primary purpose of the Bellman-Ford algorithm. While detecting cycles in directed graphs is a feature of the algorithm, it serves as a byproduct of its primary function rather than its main objective. The maximum flow problem relates to network flows, which the Bellman