How to find the maximal flow in a network?

To find the maximal flow in a network, we can use the Ford-Fulkerson algorithm.

The Ford-Fulkerson algorithm is an iterative method that starts with an initial feasible flow and then repeatedly improves it until it reaches the maximum flow. The algorithm works by finding an augmenting path in the residual graph, which is a graph that represents the remaining capacity of the network after the current flow has been subtracted from the original capacities. An augmenting path is a path from the source to the sink that has positive residual capacity on all its edges. The flow along this path can be increased by the minimum residual capacity, which is the bottleneck capacity of the path.

The algorithm continues to find augmenting paths and increase the flow until no more augmenting paths can be found. At this point, the flow is maximal and cannot be increased any further. The time complexity of the Ford-Fulkerson algorithm depends on the method used to find augmenting paths, but it can be as high as O(E^2F), where E is the number of edges and F is the maximum flow.

To implement the Ford-Fulkerson algorithm, we need to represent the network as a graph with capacities on its edges. We can use an adjacency matrix or an adjacency list to store the graph, and a residual graph can be constructed from the original graph by subtracting the current flow from the capacities. The algorithm can be implemented using a depth-first search or a breadth-first search to find augmenting paths.

In summary, the Ford-Fulkerson algorithm is a method for finding the maximal flow in a network by iteratively improving an initial feasible flow. The algorithm works by finding augmenting paths in the residual graph and increasing the flow along these paths. The time complexity of the algorithm depends on the method used to find augmenting paths, but it can be as high as O(E^2F).