Ford-Fulkerson Algorithm

Algorithm Description

The Ford-Fulkerson algorithm is used to find the maximum flow in a flow network. It is based on the concept of augmenting paths, where paths with available capacity are repeatedly found from source to sink until no more augmenting paths exist.

C++ Code

#include <iostream>
#include <vector>
#include <queue>
#include <cstring>
using namespace std;

#define V 6

bool dfs(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) {
                if (v == t) {
                    parent[v] = u;
                    return true;
                }
                q.push(v);
                parent[v] = u;
                visited[v] = true;
            }
        }
    }
    return false;
}

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;

    while (dfs(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;
        }

        max_flow += path_flow;
    }

    return max_flow;
}

int main() {
    int graph[V][V] = { {0, 16, 13, 0, 0, 0},
                        {0, 0, 10, 12, 0, 0},
                        {0, 4, 0, 0, 14, 0},
                        {0, 0, 9, 0, 0, 20},
                        {0, 0, 0, 7, 0, 4},
                        {0, 0, 0, 0, 0, 0}
    };
    cout << "The maximum possible flow is " << fordFulkerson(graph, 0, 5);
    return 0;
}

Time and Space Complexity

Operation	Time Complexity	Space Complexity
Initialization	O(V^2)	O(V^2)
Ford-Fulkerson	O(E * max_flow)	O(V^2)

Where:

• V: Number of vertices in the graph.
• E: Number of edges in the graph.

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