Edmonds-Karp Algorithm is a variation of the Ford-Fulkerson algorithm used to find the maximum flow in a flow network. This algorithm improves the runtime complexity of the Ford-Fulkerson algorithm, which relies on picking an arbitrary path from source to sink, by finding a shortest path in the residual graph using breadth-first search.
The algorithm works as follows:
Initialize the flow network with capacities, a source s and a sink t.
Initialize the flow f to 0.
While there exists a path p from s to t in the residual network:
Find the minimum capacity residual edge c(p) along path p.
Augment flow f along path p by c(p).
Update the residual capacities of edges on path p.
Return the maximum flow f.
Here is an example:
Consider a flow network with 4 nodes and 5 edges. The capacities of the edges are given as follows:
The source is node s and the sink is node t. We aim to find the maximum flow in the network.
We begin by initializing the flow to 0. We then find the shortest path from s to t in the residual network using BFS, which is:
s -> 2 -> 4 -> t
The minimum capacity residual edge along this path is 4, so we augment the flow by 4. The updated flow network becomes:
We repeat this process until there are no more paths from s to t in the residual network. The final maximum flow is 23.
Answer: The Edmonds-Karp algorithm is a variant of the Ford-Fulkerson algorithm for computing the maximum flow in a network. The main difference between the two algorithms is that Edmonds-Karp uses BFS (Breadth-First Search) to find the augmenting path, while Ford-Fulkerson uses DFS (Depth-First Search).
Answer: The time complexity of Edmonds-Karp algorithm is O(V E^2), where V is the number of vertices and E is the number of edges in the graph.
Answer: The Edmonds-Karp algorithm does not directly handle multiple sources and sinks in a network flow problem. To handle such cases, we can introduce a new source vertex s’ and a new sink vertex t’, and connect all the original sources to s’ with infinite capacity edges, and all the original sinks to t’ with infinite capacity edges. Then, we can run the Edmonds-Karp algorithm on the modified graph to find the maximum flow.
Answer: No, the Edmonds-Karp algorithm cannot handle negative edge weights in a network flow problem. If there are negative edge weights, we can use the Bellman-Ford algorithm to detect negative cycles and then eliminate them or report that the problem has no well-defined solution.
Answer: The Edmonds-Karp algorithm terminates because it always finds a shortest augmenting path (in terms of the number of edges) from the source to the sink in each iteration. Since the capacity of the augmenting path is strictly positive, the total flow increases by at least one unit in each iteration, until there is no more augmenting path. Therefore, the algorithm always terminates and finds the maximum flow in a network.