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Table of contents
1.
Problem
2.
Algorithm
2.1.
Implementation
2.2.
Time Complexity
2.3.
Space Complexity
3.
FAQs
4.
Key Takeaways 
Last Updated: Mar 27, 2024

Find minimum S-T cut in a flow network

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Ashwin Goyal
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18 Jun, 2024 @ 01:30 PM

Problem

You are given a multigraph with no self-loops. Let us represent the graph by G(V, E) where V is the set of vertices and E is the multiset of edges. 

Cut: A cut in this graph is defined as the partition of vertices into two non-empty sets (S, V-S). (Note: S != V and S != ɸ).

Weight of a cut: The weight of a cut is defined as the sum of weights of edges crossing the cut.

S-T cut: A S-T cut is a cut that divides the source and the sink into different subsets. The weight of an S-T cut is the total weight of edges going from the source’s set to the sink’s set.

Minimum S-T cut: A S-T cut having the minimum weight is called a minimum cut.

You have to find a minimum S-T cut of graph G. You have to divide the vertices into two non-empty sets, one containing source and another containing sink, such that the weight of cross edges is minimized.

Algorithm

Prerequisite: Ford-Fulkerson Algorithm

We use the Ford-Fulkerson algorithm to find maximum flow from source to sink in a graph. In the Ford-Fulkerson algorithm, we study that the maximum flow through a path equals the minimum capacity edge on that path.

For example, consider the following image.

The maximum flow from S to T in the above image is 3.

Explanation:

  1. Choose the path S->A->T. The minimum capacity edge in this path is A-T having a capacity equal to 1. Therefore, a maximum 1 unit flow can pass through this edge. Now, we subtract this value from all the edges in this path. The capacity of S->A will become 1 and A->T will become 0.
  2. Choose the path S->A->B->T. The minimum capacity edge here is S->A and A->B, having capacity 1. Now we subtract this value from all the edges of this path. 
  3. Now we can choose the path from S->B->T. Again, the maximum amount of flow that can pass through this edge equals 1. 

Thus, from the Ford-Fulkerson algorithm, we can get the capacity of the minimum S-T cut. The thing left to do is find all the edges that form the minimum S-T cut. We can find the edges using the final residual graph of the Ford-Fulkerson algorithm.

Let V be the set of vertices that are reachable from the source in the residual graph. This means that these vertices belong to the same set as S. All the remaining vertices belong to the other set as T. Now the edges of the minimum S-T cut will be all the edges from the first set to the second set. 

Implementation

#include <bits/stdc++.h>
using namespace std;
 
// Function to perform BFS
int bfs(vector<vector<int>>residualGraph, int s, int t, vector<int>&parent){
    int n = residualGraph.size();
    vector<int>vis(n, 0);

    // Initialising queue and visited array
    queue<int>q;
    vis[s]=1;
    parent[s]=-1;
    q.push(s);
 
    while(!q.empty()){
        int u = q.front();
        q.pop();
        for(int v=0; v<n; v++){

            // If node is not visited and capacity is greater than 0
            if (!vis[v] && residualGraph[u][v]>0){
                q.push(v);
                parent[v]=u;
                vis[v]=1;
            }
        }
    }

    // Will return 1 only if T was reachable from S
    return (vis[t]==1);
 
}

// Function to perform DFS
void dfs(vector<vector<int>>residualGraph, int s, vector<int>&vis){
    vis[s]=1;
    int n = residualGraph.size();
    for(int i=0; i<n; i++){
        if (residualGraph[s][i] && vis[i]==0){
            dfs(residualGraph, i, vis);
        }
    }
}

// Function to find minimum S-T cut of the given graph
void solve(vector<vector<int>>graph, int s, int t){

    // Initialising residual graph
    vector<vector<int>>residualGraph;
    int n = graph.size();
    for(int i=0; i<n; i++){
        residualGraph.push_back(graph[i]);
    }
    vector<int>parent(n);

    // Performing BFS till source and sink are reachable
    while(bfs(residualGraph, s, t, parent)){

        // Iterating over the path from sink to source
        int cur_flow = INT_MAX;
        for (int i=t; i!=s; i=parent[i]){
            int j = parent[i];

            // Updating the value of current flow through path
            cur_flow = min(cur_flow, residualGraph[j][i]);
        }

        // Updating the residual graph
        for(int i=t; i!=s; i=parent[i]){
            int j = parent[i];
            residualGraph[j][i] -= cur_flow;
            residualGraph[i][j] += cur_flow;
        }
    }
   
    // Till now we have got the weight of minimum S-T cut
    vector<int>vis(n, 0);

    // Performing DFS to see the vertices that are reachable from S
    dfs(residualGraph, s, vis);
   
    // Printing the edges if the minimum S-T cut
    for (int i=0; i<n;i++){
        for (int j=0; j<n; j++){
            if (vis[i] && !vis[j] && graph[i][j]){
                cout<<i<<" -> "<<j<<endl;
            }
        }
    }
    return;    
}
 
// Driver code
int main(){
    vector<vector<int>>graph;
    int s=0, t=5;
    graph.push_back({0, 10, 14, 4, 0, 0});
    graph.push_back({0, 0, 8, 12, 0, 0});
    graph.push_back({0, 6, 0, 0, 12, 0});
    graph.push_back({0, 10, 6, 0, 0, 20});
    graph.push_back({0, 8, 0, 6, 0, 4});
    graph.push_back({0, 0, 0, 0, 0, 0});
    solve(graph, s, t);
}

Output

3 -> 5
4 -> 5

Time Complexity

In the above implementation, to find the minimum S-T cut, we use BFS to find the path from S to T having the minimum number of edges. For every such path, we are finding the edge with minimum capacity in this path to find the maxflow from it. Since we use an adjacency matrix representation, BFS will take O(V^2) time, where V is the number of vertices.

We perform the BFS until T is not reachable from S. In every iteration, we pick one edge from the path and make its capacity 0. Therefore, the maximum number of times we perform BFS will be O(E), where E is the number of edges.

Therefore, the time complexity of the above approach is O(V^3 * E).

Space Complexity

We used an adjacency matrix representation of the graph, which will take O(V^2) space. Other than this, we created parent and visited arrays that will take O(V) space. Therefore, the total space complexity of the above approach is O(V^2).

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FAQs

  1. What is the Ford-Fulkerson Algorithm?
    The Ford-Fulkerson algorithm is a greedy algorithm to find the maximum flow in a graph from source to sink. This algorithm chooses a path from the source to a sink and sends some flow through this path. We repeat this process until the source and sink are unreachable.
  2. What is the Edmond-Karp algorithm?
    Edmon-Karp algorithm is an implementation of the Ford-Fulkerson algorithm to compute the maximum flow in a graph. This algorithm uses BFS to find the shortest path from the source to the sink and send some flow through this path. Edmond-Karp algorithm works in O(E^2 * V) time.
  3. How to find the minimum S-T cut in a flow network?
    We use the Ford-Fulkerson algorithm to find the weight of the minimum S-T cut in a flow network. The edges of the min-cut are found using the final residual graph. All the vertices that are reachable from the source form one set and the remaining form another set. The edges from the first set to the second form the minimum S-T cut.

Key Takeaways 

In this article, we learned how to find the minimum S-T cut in a flow network using the Ford-Fulkerson algorithm. We also saw its implementation in C++. We hope that this blog has helped you enhance your knowledge of the algorithm and if you would like to learn more, check out our articles on Coding Ninjas Studio. Do upvote our blog to help other ninjas grow. Happy Coding!

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