ford fulkerson maximum flow
// C++ program for implementation of Ford Fulkerson
// algorithm
#include <iostream>
#include <limits.h>
#include <queue>
#include <string.h>
using namespace std;
// Number of vertices in given graph
#define V 6
/* Returns true if there is a path from source 's' to sink
't' in residual graph. Also fills parent[] to store the
path */
bool bfs(int rGraph[V][V], int s, int t, int parent[])
{
// Create a visited array and mark all vertices as not
// visited
bool visited[V];
memset(visited, 0, sizeof(visited));
// Create a queue, enqueue source vertex and mark source
// vertex as visited
queue<int> q;
q.push(s);
visited[s] = true;
parent[s] = -1;
// Standard BFS Loop
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 we find a connection to the sink node,
// then there is no point in BFS anymore We
// just have to set its parent and can return
// true
if (v == t) {
parent[v] = u;
return true;
}
q.push(v);
parent[v] = u;
visited[v] = true;
}
}
}
// We didn't reach sink in BFS starting from source, so
// return false
return false;
}
// Returns the maximum flow from s to t in the given graph
int fordFulkerson(int graph[V][V], int s, int t)
{
int u, v;
// Create a residual graph and fill the residual graph
// with given capacities in the original graph as
// residual capacities in residual graph
int rGraph[V]
[V]; // Residual graph where rGraph[i][j]
// indicates residual capacity of edge
// from i to j (if there is an edge. If
// rGraph[i][j] is 0, then there is not)
for (u = 0; u < V; u++)
for (v = 0; v < V; v++)
rGraph[u][v] = graph[u][v];
int parent[V]; // This array is filled by BFS and to
// store path
int max_flow = 0; // There is no flow initially
// Augment the flow while tere is path from source to
// sink
while (bfs(rGraph, s, t, parent)) {
// Find minimum residual capacity of the edges along
// the path filled by BFS. Or we can say find the
// maximum flow through the path found.
int path_flow = INT_MAX;
for (v = t; v != s; v = parent[v]) {
u = parent[v];
path_flow = min(path_flow, rGraph[u][v]);
}
// update residual capacities of the edges and
// reverse edges along the path
for (v = t; v != s; v = parent[v]) {
u = parent[v];
rGraph[u][v] -= path_flow;
rGraph[v][u] += path_flow;
}
// Add path flow to overall flow
max_flow += path_flow;
}
// Return the overall flow
return max_flow;
}
// Driver program to test above functions
int main()
{
// Let us create a graph shown in the above example
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;
}