Algorithm Design Questions
The Bellman-Ford algorithm is used to find the shortest path in a graph that may contain negative edge weights. It works by iteratively relaxing the edges of the graph until the shortest path distances are determined.
The algorithm starts by initializing the distance of the source vertex to 0 and the distance of all other vertices to infinity. Then, it iterates through all the edges of the graph V-1 times, where V is the number of vertices. In each iteration, it relaxes the edges by updating the distance of the destination vertex if a shorter path is found.
During each iteration, the algorithm checks all the edges and updates the distance of the destination vertex if the sum of the distance from the source vertex to the current vertex and the weight of the edge is smaller than the current distance of the destination vertex. This process is repeated until no further updates are possible.
If after V-1 iterations, there are still updates being made, it indicates the presence of a negative cycle in the graph. A negative cycle is a cycle whose total weight is negative, and it can cause the shortest path to be undefined.
The Bellman-Ford algorithm has a time complexity of O(V * E), where V is the number of vertices and E is the number of edges in the graph. It is slower than Dijkstra's algorithm but can handle graphs with negative edge weights.