What is Bellman-Ford algorithm for finding the shortest path?

The Bellman-Ford algorithm is a method for finding the shortest path in a weighted graph.

The Bellman-Ford algorithm is used to find the shortest path between a source vertex and all other vertices in a weighted graph. It works by relaxing each edge in the graph repeatedly, which means updating the distance to each vertex based on the distance to its neighbours. The algorithm starts by setting the distance to the source vertex to 0 and the distance to all other vertices to infinity. Then, it relaxes each edge in the graph V-1 times, where V is the number of vertices in the graph. After each iteration, the algorithm updates the distance to each vertex if a shorter path has been found.

If the graph contains a negative weight cycle, the algorithm will detect it and report that no shortest path exists. A negative weight cycle is a cycle in the graph where the sum of the weights is negative. This is because in such a cycle, the algorithm can keep reducing the distance to a vertex indefinitely by repeatedly traversing the cycle.

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 in the graph. This makes it less efficient than other algorithms such as Dijkstra's algorithm, which has a time complexity of O(E log V) but cannot handle negative weight edges.