Minimize the number of weakly connected nodes

Difficulty Level : Hard
Last Updated : 31 Dec, 2020

Given an undirected graph, task is to find the minimum number of weakly connected nodes after converting this graph into directed one.

Weakly Connected Nodes : Nodes which are having 0 indegree(number of incoming edges).

Prerequisite : BFS traversal

Examples :

Input : 4 4 
        0 1
        1 2
        2 3
        3 0
Output : 0 disconnected components

Input : 6 5
       1 2
       2 3
       4 5
       4 6
       5 6
Output : 1 disconnected components

Explanation :

Approach : We find a node which helps in traversing maximum nodes in a single walk. To cover all possible paths, DFS graph traversal technique is used for this.

Do the above steps to traverse the graph. Now, iterate through graph again and check which nodes are having 0 indegree.

C++

// C++ code to minimize the number
// of weakly connected nodes
#include <bits/stdc++.h>
using namespace std;
 
// Set of nodes which are traversed
// in each launch of the DFS
set<int> node;
vector<int> Graph[10001];
 
// Function traversing the graph using DFS
// approach and updating the set of nodes
void dfs(bool visit[], int src)
{
    visit[src] = true;
    node.insert(src);
    int len = Graph[src].size();
    for (int i = 0; i < len; i++)   
        if (!visit[Graph[src][i]])       
            dfs(visit, Graph[src][i]);
}
 
// building a undirected graph
void buildGraph(int x[], int y[], int len){
 
    for (int i = 0; i < len; i++)
    {
        int p = x[i];
        int q = y[i];
        Graph[p].push_back(q);
        Graph[q].push_back(p);
    }
}
 
// computes the minimum number of disconnected
// components when a bi-directed graph is
// converted to a undirected graph
int compute(int n)
{
    // Declaring and initializing
    // a visited array
    bool visit[n + 5];
    memset(visit, false, sizeof(visit));
    int number_of_nodes = 0;
 
    // We check if each node is
    // visited once or not
    for (int i = 0; i < n; i++)
    {
        // We only launch DFS from a
        // node iff it is unvisited.
        if (!visit[i]) {
 
            // Clearing the set of nodes
            // on every relaunch of DFS
            node.clear();
             
            // relaunching DFS from an
            // unvisited node.
            dfs(visit, i);
            
            // iterating over the node set to count the
            // number of nodes visited after making the
            // graph directed and storing it in the
            // variable count. If count / 2 == number
            // of nodes - 1, then increment count by 1.
            int count = 0;        
            for (auto it = node.begin(); it != node.end(); ++it)
                count += Graph[(*it)].size();
         
            count /= 2;       
            if (count == node.size() - 1)
               number_of_nodes++;
        }
    }
    return number_of_nodes;
}
 
//Driver function
int main()
{
    int n = 6,m = 4;
    int x[m + 5] = {1, 1, 4, 4};
    int y[m+5] = {2, 3, 5, 6};
     
    /*For given x and y above, graph is as below :
        1-----2         4------5
        |               |
        |               |
        |               |
        3               6
         
        // Note : This code will work for
        // connected graph also as :
        1-----2
        |     | \
        |     |  \
        |     |   \
        3-----4----5
    */
     
    // Building graph in the form of a adjacency list
    buildGraph(x, y, n);
    cout << compute(n) << " weakly connected nodes";
     
    return 0;
}

Python3

# Python3 code to minimize the number
# of weakly connected nodes
  
# Set of nodes which are traversed
# in each launch of the DFS
node = set()
Graph = [[] for i in range(10001)]
  
# Function traversing the graph using DFS
# approach and updating the set of nodes
def dfs(visit, src):
 
    visit[src] = True
    node.add(src)
    llen = len(Graph[src])
     
    for i in range(llen):
        if (not visit[Graph[src][i]]):
            dfs(visit, Graph[src][i])
 
# Building a undirected graph
def buildGraph(x, y, llen):
     
    for i in range(llen):
        p = x[i]
        q = y[i]
        Graph[p].append(q)
        Graph[q].append(p)
     
# Computes the minimum number of disconnected
# components when a bi-directed graph is
# converted to a undirected graph
def compute(n):
 
    # Declaring and initializing
    # a visited array
    visit = [False for i in range(n + 5)]
     
    number_of_nodes = 0
  
    # We check if each node is
    # visited once or not
    for i in range(n):
         
        # We only launch DFS from a
        # node iff it is unvisited.
        if (not visit[i]):
  
            # Clearing the set of nodes
            # on every relaunch of DFS
            node.clear()
              
            # Relaunching DFS from an
            # unvisited node.
            dfs(visit, i)
             
            # Iterating over the node set to count the
            # number of nodes visited after making the
            # graph directed and storing it in the
            # variable count. If count / 2 == number
            # of nodes - 1, then increment count by 1.
            count = 0     
             
            for it in node:
                count += len(Graph[(it)])
          
            count //= 2
             
            if (count == len(node) - 1):
               number_of_nodes += 1
         
    return number_of_nodes
 
# Driver code
if __name__=='__main__':
     
    n = 6
    m = 4
    x = [ 1, 1, 4, 4, 0, 0, 0, 0, 0 ]
    y = [ 2, 3, 5, 6, 0, 0, 0, 0, 0 ]
      
    '''For given x and y above, graph is as below :
        1-----2         4------5
        |               |
        |               |
        |               |
        3               6
          
        # Note : This code will work for
        # connected graph also as :
        1-----2
        |     | \
        |     |  \
        |     |   \
        3-----4----5
    '''
      
    # Building graph in the form of a adjacency list
    buildGraph(x, y, n)
     
    print(str(compute(n)) +
          " weakly connected nodes")
      
# This code is contributed by rutvik_56

C#

// C# code to minimize the number
// of weakly connected nodes
using System;
using System.Collections;
using System.Collections.Generic;
 
class GFG{
     
// Set of nodes which are traversed
// in each launch of the DFS
static HashSet<int> node = new HashSet<int>();
static List<int> []Graph = new List<int>[10001];
  
// Function traversing the graph using DFS
// approach and updating the set of nodes
static void dfs(bool []visit, int src)
{
    visit[src] = true;
    node.Add(src);
    int len = Graph[src].Count;
     
    for(int i = 0; i < len; i++)   
        if (!visit[Graph[src][i]])       
            dfs(visit, Graph[src][i]);
}
  
// Building a undirected graph
static void buildGraph(int []x, int []y, int len)
{
    for(int i = 0; i < len; i++)
    {
        int p = x[i];
        int q = y[i];
        Graph[p].Add(q);
        Graph[q].Add(p);
    }
}
  
// Computes the minimum number of disconnected
// components when a bi-directed graph is
// converted to a undirected graph
static int compute(int n)
{
     
    // Declaring and initializing
    // a visited array
    bool []visit = new bool[n + 5];
    Array.Fill(visit, false);
 
    int number_of_nodes = 0;
  
    // We check if each node is
    // visited once or not
    for(int i = 0; i < n; i++)
    {
         
        // We only launch DFS from a
        // node iff it is unvisited.
        if (!visit[i])
        {
             
            // Clearing the set of nodes
            // on every relaunch of DFS
            node.Clear();
              
            // Relaunching DFS from an
            // unvisited node.
            dfs(visit, i);
             
            // Iterating over the node set to count the
            // number of nodes visited after making the
            // graph directed and storing it in the
            // variable count. If count / 2 == number
            // of nodes - 1, then increment count by 1.
            int count = 0;        
            foreach(int it in node)
                count += Graph[(it)].Count;
          
            count /= 2;
             
            if (count == node.Count - 1)
               number_of_nodes++;
        }
    }
    return number_of_nodes;
}
  
// Driver Code
static void Main(string []args)
{
    int n = 6;
    for(int i = 0; i < 10001; i++)
    {
        Graph[i] = new List<int>();
    }
    int []x = { 1, 1, 4, 4, 0, 0, 0, 0 };
    int []y = { 2, 3, 5, 6, 0, 0, 0, 0 };
      
    /*For given x and y above, graph is as below :
        1-----2         4------5
        |               |
        |               |
        |               |
        3               6
          
        // Note : This code will work for
        // connected graph also as :
        1-----2
        |     | \
        |     |  \
        |     |   \
        3-----4----5
    */
      
    // Building graph in the form of a adjacency list
    buildGraph(x, y, n);
    Console.Write(compute(n) +
    " weakly connected nodes");
}
}
 
// This code is contributed by pratham76

Output:

2 weakly connected nodes

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