Ford-Fulkerson Algorithm

The Ford-Fulkerson algorithm is a graph algorithm used to compute the maximum flow in a network. It is an iterative approach that finds augmenting paths in the residual graph until no more augmenting paths exist.

The algorithm is based on the concept of residual graph, which represents the remaining capacity of edges in the original graph. Initially, the residual graph is the same as the original graph.

The Ford-Fulkerson algorithm repeatedly performs a breadth-first search (BFS) on the residual graph to find an augmenting path from the source to the sink. An augmenting path is a path in the residual graph that allows for additional flow to be sent from the source to the sink.

Once an augmenting path is found, the algorithm updates the flow along the path and updates the residual capacities of the edges in the residual graph. This process is repeated until no more augmenting paths exist.

The maximum flow in the network is determined by summing the flow along all the edges leaving the source node.

The Java code snippet below demonstrates the implementation of the Ford-Fulkerson algorithm:

1class FordFulkerson {
2
3    public int fordFulkerson(int[][] graph, int source, int sink) {
4        // Implementation...
5    }
6
7    private boolean bfs(int[][] graph, int source, int sink, int[] parent) {
8        // Implementation...
9    }
10
11    public static void main(String[] args) {
12        FordFulkerson fordFulkerson = new FordFulkerson();
13
14        int[][] graph = {
15            // Input graph...
16        };
17        int source = // Source node;
18        int sink = // Sink node;
19        int maxFlow = fordFulkerson.fordFulkerson(graph, source, sink);
20        System.out.println("Maximum Flow: " + maxFlow);
21    }
22}

In the example code above, we have a FordFulkerson class that implements the Ford-Fulkerson algorithm. The fordFulkerson method takes the input graph, source node, and sink node as parameters and returns the maximum flow in the network.

The method uses a BFS function, bfs, to find a path from the source node to the sink node in the residual graph. The bfs function uses a boolean array to keep track of visited nodes and a queue to perform the BFS traversal.

The main method demonstrates the usage of the FordFulkerson class by creating an example input graph, setting the source and sink nodes, and calling the fordFulkerson method to compute the maximum flow. Finally, the maximum flow is printed to the console.

The Ford-Fulkerson algorithm is an important algorithm in the field of network flows and has various applications, such as in traffic routing, network planning, and resource allocation. Its time complexity can vary depending on the implementation and the specific graph structure, but it is generally considered efficient for many practical scenarios.

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}

```java

public class FordFulkerson {

​

    public int fordFulkerson(int[][] graph, int source, int sink) {

        int[][] residualGraph = new int[graph.length][graph[0].length];

        for (int i = 0; i < graph.length; i++) {

            for (int j = 0; j < graph[0].length; j++) {

                residualGraph[i][j] = graph[i][j];

            }

        }

        int[] parent = new int[graph.length];

        int maxFlow = 0;

​

        while (bfs(residualGraph, source, sink, parent)) {

            int pathFlow = Integer.MAX_VALUE;

​

            for (int v = sink; v != source; v = parent[v]) {

                int u = parent[v];

                pathFlow = Math.min(pathFlow, residualGraph[u][v]);

            }

​

            for (int v = sink; v != source; v = parent[v]) {

                int u = parent[v];

                residualGraph[u][v] -= pathFlow;

                residualGraph[v][u] += pathFlow;

            }

​

            maxFlow += pathFlow;

        }