Graph Theory Questions Medium
The Edmonds-Karp algorithm is a variation of the Ford-Fulkerson algorithm used to solve the maximum flow problem in a graph. It is an efficient algorithm that guarantees to find the maximum flow in a network.
The algorithm starts by initializing the flow in all edges to zero. Then, it repeatedly finds an augmenting path from the source to the sink using a breadth-first search (BFS) approach. An augmenting path is a path in the residual graph that has available capacity for increasing the flow.
During each iteration, the algorithm finds the shortest augmenting path in terms of the number of edges using BFS. This ensures that the algorithm runs in O(VE^2) time complexity, where V is the number of vertices and E is the number of edges in the graph.
Once an augmenting path is found, the algorithm determines the maximum amount of flow that can be pushed through this path, which is the minimum capacity of all edges in the path. This value is called the bottleneck capacity.
Then, the algorithm updates the flow along the augmenting path by increasing the flow in the forward edges and decreasing the flow in the backward edges by the bottleneck capacity. This process is repeated until no more augmenting paths can be found.
Finally, the algorithm outputs the maximum flow, which is the sum of the flow values leaving the source vertex.
The Edmonds-Karp algorithm guarantees to find the maximum flow in a graph and has a better worst-case time complexity compared to the original Ford-Fulkerson algorithm.