BellmanFords

The Bellman-Ford algorithm is a computer algorithm that computes shortest paths from a single vertex to all other vertices in a weighted digraph. It is capable of handling graphs with negative edge weights, which distinguishes it from Dijkstra's algorithm. The algorithm was developed by Richard Bellman and Lester Ford Jr. in the late 1950s.

The Bellman-Ford algorithm works by repeatedly relaxing all edges in the graph. Relaxation is the process of checking if the path to a vertex can be shortened by going through another vertex. The algorithm performs this relaxation process V-1 times, where V is the number of vertices in the graph. After V-1 iterations, if the graph contains no negative-weight cycles reachable from the source, the shortest paths are guaranteed to have been found.

A key feature of the Bellman-Ford algorithm is its ability to detect negative-weight cycles. If, after V-1 iterations, any edge can still be relaxed, it indicates the presence of a negative-weight cycle. This is because a shortest path in a graph without negative cycles can have at most V-1 edges. If a path can be improved after V-1 relaxations, it implies that the path is traversing a cycle with a negative total weight, allowing for infinitely short paths.

The time complexity of the Bellman-Ford algorithm is O(VE), where V is the number of vertices and E is the number of edges. This is less efficient than Dijkstra's algorithm (O(E + V log V) with a Fibonacci heap) for graphs without negative edge weights. However, its ability to handle negative weights makes it essential for certain applications, such as in routing protocols like the distance-vector protocol where negative weights can arise.