Find minimum s-t cut in a flow network in C++

Suppose we have following flow network. As we know an s-t cut is a cut that requires the source s node and a sink t node to be in different subsets, and it includes edges going from the source set to the sink side. Here the capacity of an s-t cut is represented by the sum of each edge capacity in the cut-set. Here we have to find minimum capacity s-t cut of the given network. Here the expected output is all edges of the minimum cut.

So, if the input is like

then the output will be [(1,3), (4,3), (4,5)]

To solve this, we will follow these steps −

• NODES = 6

• Define a function bfs(), this will take graph, src, sink, array par,

• Define an array vis of size − NODES. and fill with 0

• Define one queue que

• insert src into que

• vis[src] := true and par[src] := -1

• while (que is not empty), do −

  • u1 := first element of que

  • delete element from que

  • for initialize v1 := 0, when v1 < NODES, update (increase v1 by 1), do−

    • if vis[v1] is false and graph[u1, v1] > 0, then −

      • insert v1 into que

      • par[v1] := u1

      • vis[v1] := true

• return true when vis[sink] is true

• Define a function dfs(), this will take graph, src, array vis,

• vis[src] := true

• for initialize i := 0, when i < NODES, update (increase i by 1), do−

  • if graph[src, i] is non-zero and vis[i] is false, then −

    • dfs(graph, i, vis)

• From the main method, do the following −

• Define an array temp_graph and copy graph into it

• Define an array par of size: NODES.

• while bfs(temp_graph, src, sink, par) is true, do −

  • path_flow := inf

  • for initialize v := sink, when v is not equal to src, update v:=par[v], do −

    • u := par[v]

    • path_flow := minimum of path_flow and temp_graph[u, v]

  • for initialize v := sink, when v is not equal to src, update v:=par[v], do −

    • u := par[v]

    • temp_graph[u, v] := temp_graph[u, v] - path_flow

    • temp_graph[v, u] := temp_graph[v, u] + path_flow

• Define an array vis of size − NODES. and fill with false

• dfs(temp_graph, src, vis)

• for initialize i := 0, when i − NODES, update (increase i by 1), do −

• for initialize j := 0, when j − NODES, update (increase j by 1), do −

  • if vis[i] is non-zero and vis[j] is false and graph[i, j] is non-zero, then −

    • display (i, j) as edge

  • return

Example (C++)

Let us see the following implementation to get better understanding −

 Live Demo

#include <bits/stdc++.h>
using namespace std;
#define NODES 6
int bfs(int graph[NODES][NODES], int src, int sink, int par[]) {
   bool vis[NODES];
   memset(vis, 0, sizeof(vis));
   queue <int> que;
   que.push(src);
   vis[src] = true;
   par[src] = -1;
   while (!que.empty()) {
      int u1 = que.front();
      que.pop();
      for (int v1=0; v1<NODES; v1++){
         if (vis[v1]==false && graph[u1][v1] > 0) {
            que.push(v1);
            par[v1] = u1;
            vis[v1] = true;
         }
      }
   }
   return (vis[sink] == true);
}
void dfs(int graph[NODES][NODES], int src, bool vis[]) {
   vis[src] = true;
   for (int i = 0; i < NODES; i++)
   if (graph[src][i] && !vis[i])
   dfs(graph, i, vis);
}
void minCut(int graph[NODES][NODES], int src, int sink) {
   int u, v;
   int temp_graph[NODES][NODES];
   for (u = 0; u < NODES; u++)
      for (v = 0; v < NODES; v++)
         temp_graph[u][v] = graph[u][v];
   int par[NODES];
   while (bfs(temp_graph, src, sink, par)){
      int path_flow = INT_MAX;
      for (v=sink; v!=src; v=par[v]) {
         u = par[v];
         path_flow = min(path_flow, temp_graph[u][v]);
      }
      for (v=sink; v != src; v=par[v]) {
         u = par[v];
         temp_graph[u][v] -= path_flow;
         temp_graph[v][u] += path_flow;
      }
   }
   bool vis[NODES];
   memset(vis, false, sizeof(vis));
   dfs(temp_graph, src, vis);
   for (int i = 0; i < NODES; i++)
      for (int j = 0; j < NODES; j++)
         if (vis[i] && !vis[j] && graph[i][j])
            cout << "("<< i << ", " << j << ")" << endl;
   return;
}
int main() {
   int graph1[NODES][NODES] = {
      {0, 17, 14, 0, 0, 0},
      {0, 0, 11, 13, 0, 0},
      {0, 5, 0, 0, 15, 0},
      {0, 0, 9, 0, 0, 21},
      {0, 0, 0, 8, 0, 5},
      {0, 0, 0, 0, 0, 0}
   };
   minCut(graph1, 0, 5);
}

Input

{{0, 17, 14, 0, 0, 0},
{0, 0, 11, 13, 0, 0},
{0, 5, 0, 0, 15, 0},
{0, 0, 9, 0, 0, 21
{0, 0, 0, 8, 0, 5},
{0, 0, 0, 0, 0, 0}};

Output

(1, 3)
(4, 3)
(4, 5)