Edmonds-Karp Algorithm
The Edmonds-Karp algorithm is an optimization of the Ford-Fulkerson algorithm for finding the maximum flow in a network. It improves the efficiency of the Ford-Fulkerson algorithm by using BFS instead of DFS to find augmenting paths.
The algorithm follows a similar approach to the Ford-Fulkerson algorithm. Starting with an initial flow of zero, it iteratively finds augmenting paths in the residual graph using BFS. An augmenting path is a path from the source to the sink in the residual graph where each edge has a positive residual capacity.
Unlike the Ford-Fulkerson algorithm, which uses DFS to find augmenting paths, the Edmonds-Karp algorithm uses BFS. This choice of BFS ensures that the shortest augmenting path is found in each iteration, guaranteeing efficiency.
The Java code snippet below demonstrates the implementation of the Edmonds-Karp algorithm:
xxxxxxxxxxLet's continue on the next screen!import java.util.*;
public class EdmondsKarp {
xxxxxxxxxx}private static final int INF = 1000000000;public int edmondsKarp(int[][] graph, int source, int sink) { int n = graph.length; int[][] residualGraph = new int[n][n]; // Initialize the residual graph for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { residualGraph[i][j] = graph[i][j]; } } int maxFlow = 0; // Find augmenting paths using BFS int[] parent = new int[n]; while (bfs(residualGraph, source, sink, parent)) { int pathFlow = INF; // Find the minimum residual capacity along the augmenting path for (int v = sink; v != source; v = parent[v]) { int u = parent[v]; pathFlow = Math.min(pathFlow, residualGraph[u][v]); } // Update the residual capacities and reverse edges along the augmenting path for (int v = sink; v != source; v = parent[v]) {}`
xxxxxxxxxxLet's continue on the next screen!