Graph Theory Questions Medium
The Bellman-Ford algorithm is a popular algorithm used to find the shortest path from a source vertex to all other vertices in a weighted graph. It can handle graphs with negative edge weights, unlike Dijkstra's algorithm.
The algorithm works by iteratively relaxing the edges of the graph. It starts by initializing the distance of the source vertex to 0 and the distance of all other vertices to infinity. Then, it performs a series of relaxation steps, where it considers each edge in the graph and updates the distance of the destination vertex if a shorter path is found.
The relaxation step involves comparing the current distance of the destination vertex with the sum of the distance of the source vertex and the weight of the edge connecting them. If the sum is smaller, the distance of the destination vertex is updated to the new value. This process is repeated for all edges in the graph for a total of V-1 iterations, where V is the number of vertices in the graph.
After V-1 iterations, the algorithm checks for negative cycles in the graph. If a shorter path is found during the Vth iteration, it means that there is a negative cycle in the graph, and the algorithm cannot find a shortest path. Otherwise, the algorithm has successfully found the shortest path from the source vertex to all other vertices.
The time complexity of the Bellman-Ford algorithm is O(V * E), where V is the number of vertices and E is the number of edges in the graph.