In this tutorial, you will learn what the **Ford-Fulkerson algorithm** is. Likewise, you will discover working instances of discovering maximum flow in a flow network in C, C++, Java, and Python.

Ford-Fulkerson algorithm is a greedy approach for calculating the maximum conceivable flow in a network or a graph.

A term, **flow network**, is used to depict a network of vertices and edges with a source (S) and a sink (T). Every vertex, with the exception of S and T, can get and send an equivalent measure of stuff through it. S can just send and T can just get stuff.

We can envision the comprehension of the algorithm using a flow of liquid inside a network of pipes of various capacities. Each pipe has a specific capacity of liquid it can move at a case. For this algorithm, we will discover how much liquid can be moved from the source to the sink at an instance using the network.

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## Terminologies Used

**Augmenting Path**

It is the way accessible in a flow network.

**Residual Graph**

It addresses the flow network that has extra conceivable flow.

**Residual Capacity**

It is the capacity of the edge in the wake of taking away the flow from the maximum capacity.

## How does Ford-Fulkerson Algorithm work?

The algorithm follows:

- Introduce the flow in all the edges to 0.

2. While there is an augmenting path between the source and the sink, add this way to the flow.

3. Update the residual graph.

We can likewise consider reverse-path whenever required since, in such a case that we don’t think about them, we may never find a maximum flow.

The above ideas can be perceived with the example beneath.

## Ford-Fulkerson Example

The flow of the multitude of edges is 0 toward the start.

1. Select any arbitrary path from S to T. In this step, we have selected path S-A-B-T.

The minimum capacity among the three edges is 2 (B-T). In view of this, update the flow/capacity in every way.

2. Select another way S-D-C-T. The minimum capacity among these edges is 3 (S-D).

Update the capacities according to this.

3. Presently, let us consider the reverse way B-D too. Choosing way S-A-B-D-C-T. The minimum residual capacity among the edges is 1 (D-C).

Updating the capacities.

The capacity for forward and reverse ways is considered separately.

4. Adding all the flows = 2 + 3 + 1 = 6, which is the maximum conceivable flow on the flow network.

Note that assuming the capacity for any edge is full, that way can’t be used.

## Python, Java and C/C++ Examples

**Python**

# Ford-Fulkerson algorith in Python from collections import defaultdict class Graph: def __init__(self, graph): self.graph = graph self. ROW = len(graph) # Using BFS as a searching algorithm def searching_algo_BFS(self, s, t, parent): visited = [False] * (self.ROW) queue = [] queue.append(s) visited[s] = True while queue: u = queue.pop(0) for ind, val in enumerate(self.graph[u]): if visited[ind] == False and val > 0: queue.append(ind) visited[ind] = True parent[ind] = u return True if visited[t] else False # Applying fordfulkerson algorithm def ford_fulkerson(self, source, sink): parent = [-1] * (self.ROW) max_flow = 0 while self.searching_algo_BFS(source, sink, parent): path_flow = float("Inf") s = sink while(s != source): path_flow = min(path_flow, self.graph[parent[s]][s]) s = parent[s] # Adding the path flows max_flow += path_flow # Updating the residual values of edges v = sink while(v != source): u = parent[v] self.graph[u][v] -= path_flow self.graph[v][u] += path_flow v = parent[v] return max_flow graph = [[0, 8, 0, 0, 3, 0], [0, 0, 9, 0, 0, 0], [0, 0, 0, 0, 7, 2], [0, 0, 0, 0, 0, 5], [0, 0, 7, 4, 0, 0], [0, 0, 0, 0, 0, 0]] g = Graph(graph) source = 0 sink = 5 print("Max Flow: %d " % g.ford_fulkerson(source, sink))

**Java**

// Ford-Fulkerson algorith in Java import java.util.LinkedList; class FordFulkerson { static final int V = 6; // Using BFS as a searching algorithm boolean bfs(int Graph[][], int s, int t, int p[]) { boolean visited[] = new boolean[V]; for (int i = 0; i < V; ++i) visited[i] = false; LinkedList<Integer> queue = new LinkedList<Integer>(); queue.add(s); visited[s] = true; p[s] = -1; while (queue.size() != 0) { int u = queue.poll(); for (int v = 0; v < V; v++) { if (visited[v] == false && Graph[u][v] > 0) { queue.add(v); p[v] = u; visited[v] = true; } } } return (visited[t] == true); } // Applying fordfulkerson algorithm int fordFulkerson(int graph[][], int s, int t) { int u, v; int Graph[][] = new int[V][V]; for (u = 0; u < V; u++) for (v = 0; v < V; v++) Graph[u][v] = graph[u][v]; int p[] = new int[V]; int max_flow = 0; # Updating the residual calues of edges while (bfs(Graph, s, t, p)) { int path_flow = Integer.MAX_VALUE; for (v = t; v != s; v = p[v]) { u = p[v]; path_flow = Math.min(path_flow, Graph[u][v]); } for (v = t; v != s; v = p[v]) { u = p[v]; Graph[u][v] -= path_flow; Graph[v][u] += path_flow; } // Adding the path flows max_flow += path_flow; } return max_flow; } public static void main(String[] args) throws java.lang.Exception { int graph[][] = new int[][] { { 0, 8, 0, 0, 3, 0 }, { 0, 0, 9, 0, 0, 0 }, { 0, 0, 0, 0, 7, 2 }, { 0, 0, 0, 0, 0, 5 }, { 0, 0, 7, 4, 0, 0 }, { 0, 0, 0, 0, 0, 0 } }; FordFulkerson m = new FordFulkerson(); System.out.println("Max Flow: " + m.fordFulkerson(graph, 0, 5)); } }

**C**

/ Ford - Fulkerson algorith in C #include <stdio.h> #define A 0 #define B 1 #define C 2 #define MAX_NODES 1000 #define O 1000000000 int n; int e; int capacity[MAX_NODES][MAX_NODES]; int flow[MAX_NODES][MAX_NODES]; int color[MAX_NODES]; int pred[MAX_NODES]; int min(int x, int y) { return x < y ? x : y; } int head, tail; int q[MAX_NODES + 2]; void enqueue(int x) { q[tail] = x; tail++; color[x] = B; } int dequeue() { int x = q[head]; head++; color[x] = C; return x; } // Using BFS as a searching algorithm int bfs(int start, int target) { int u, v; for (u = 0; u < n; u++) { color[u] = A; } head = tail = 0; enqueue(start); pred[start] = -1; while (head != tail) { u = dequeue(); for (v = 0; v < n; v++) { if (color[v] == A && capacity[u][v] - flow[u][v] > 0) { enqueue(v); pred[v] = u; } } } return color[target] == C; } // Applying fordfulkerson algorithm int fordFulkerson(int source, int sink) { int i, j, u; int max_flow = 0; for (i = 0; i < n; i++) { for (j = 0; j < n; j++) { flow[i][j] = 0; } } // Updating the residual values of edges while (bfs(source, sink)) { int increment = O; for (u = n - 1; pred[u] >= 0; u = pred[u]) { increment = min(increment, capacity[pred[u]][u] - flow[pred[u]][u]); } for (u = n - 1; pred[u] >= 0; u = pred[u]) { flow[pred[u]][u] += increment; flow[u][pred[u]] -= increment; } // Adding the path flows max_flow += increment; } return max_flow; } int main() { for (int i = 0; i < n; i++) { for (int j = 0; j < n; j++) { capacity[i][j] = 0; } } n = 6; e = 7; capacity[0][1] = 8; capacity[0][4] = 3; capacity[1][2] = 9; capacity[2][4] = 7; capacity[2][5] = 2; capacity[3][5] = 5; capacity[4][2] = 7; capacity[4][3] = 4; int s = 0, t = 5; printf("Max Flow: %d\n", fordFulkerson(s, t)); }

**C++**

// Ford-Fulkerson algorith in C++ #include <limits.h> #include <string.h> #include <iostream> #include <queue> using namespace std; #define V 6 // Using BFS as a searching algorithm bool bfs(int rGraph[V][V], int s, int t, int parent[]) { bool visited[V]; memset(visited, 0, sizeof(visited)); queue<int> q; q.push(s); visited[s] = true; parent[s] = -1; while (!q.empty()) { int u = q.front(); q.pop(); for (int v = 0; v < V; v++) { if (visited[v] == false && rGraph[u][v] > 0) { q.push(v); parent[v] = u; visited[v] = true; } } } return (visited[t] == true); } // Applying fordfulkerson algorithm int fordFulkerson(int graph[V][V], int s, int t) { int u, v; int rGraph[V][V]; for (u = 0; u < V; u++) for (v = 0; v < V; v++) rGraph[u][v] = graph[u][v]; int parent[V]; int max_flow = 0; // Updating the residual values of edges while (bfs(rGraph, s, t, parent)) { int path_flow = INT_MAX; for (v = t; v != s; v = parent[v]) { u = parent[v]; path_flow = min(path_flow, rGraph[u][v]); } for (v = t; v != s; v = parent[v]) { u = parent[v]; rGraph[u][v] -= path_flow; rGraph[v][u] += path_flow; } // Adding the path flows max_flow += path_flow; } return max_flow; } int main() { int graph[V][V] = {{0, 8, 0, 0, 3, 0}, {0, 0, 9, 0, 0, 0}, {0, 0, 0, 0, 7, 2}, {0, 0, 0, 0, 0, 5}, {0, 0, 7, 4, 0, 0}, {0, 0, 0, 0, 0, 0}}; cout << "Max Flow: " << fordFulkerson(graph, 0, 5) << endl; }

## Ford-Fulkerson Applications

- Water distribution pipeline
- Bipartite matching problem
- Circulation with demands

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