Number of groups formed in a graph of friends

Given n friends and their friendship relations, find the total number of groups that exist. And the number of ways of new groups that can be formed consisting of people from every existing group.

If no relation is given for any person then that person has no group and singularly forms a group. If a is a friend of b and b is a friend of c, then a b and c form a group.

Examples:

Input : Number of people = 6 
        Relations : 1 - 2, 3 - 4 
                    and 5 - 6 
Output: Number of existing Groups = 3
        Number of new groups that can
        be formed = 8
Explanation: The existing groups are 
(1, 2), (3, 4), (5, 6). The new 8 groups 
that can be formed by considering a 
member of every group are (1, 3, 5), 
(1, 3, 6), (1, 4, 5), (1, 4, 6), (2, 
3, 5), (2, 3, 6), (2, 4, 5) and (2, 4,
6). 

Input:  Number of people = 6 
        Relations : 1 - 2 and 2 - 3 
Output: Number of existing Groups = 2
        Number of new groups that can
        be formed = 3
Explanation: The existing groups are 
(1, 2, 3) and (4). The new groups that 
can be formed by considering a member
of every group are (1, 4), (2, 4), (3, 4).

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

To count number of groups, we need to simply count connected components in the given undirected graph. Counting connected components can be easily done using DFS or BFS.
Since this is an undirected graph, the number of times a Depth First Search starts from an unvisited vertex for every friend is equal to the number of groups formed.

To count number of ways in which we form new groups can be done using simply formula which is (N1)*(N2)*….(Nn) where Ni is the no of people in i-th group.

// CPP program to count number of existing 
// groups and number of new groups that can 
// be formed. 
#include <bits/stdc++.h> 
using namespace std; 
  
class Graph { 
    int V; // No. of vertices 
      
    // Pointer to an array containing 
    // adjacency lists 
    list<int>* adj; 
      
    int countUtil(int v, bool visited[]);  
public: 
    Graph(int V); // Constructor 
      
    // function to add an edge to graph 
    void addRelation(int v, int w);  
    void countGroups();  
}; 
  
Graph::Graph(int V) 
{ 
    this->V = V; 
    adj = new list<int>[V]; 
} 
  
// Adds a relation as a two way edge of  
// undirected graph. 
void Graph::addRelation(int v, int w) 
{ 
    // Since indexing is 0 based, reducing 
    // edge numbers by 1. 
    v--; 
    w--; 
    adj[v].push_back(w); 
    adj[w].push_back(v);  
} 
  
// Returns count of not visited nodes reachable 
// from v using DFS. 
int Graph::countUtil(int v, bool visited[]) 
{ 
    int count = 1;  
    visited[v] = true;  
    for (auto i=adj[v].begin(); i!=adj[v].end(); ++i) 
        if (!visited[*i])  
            count = count + countUtil(*i, visited); 
    return count;         
} 
  
// A DFS based function to Count number of 
// existing groups and number of new groups 
// that can be formed using a member of 
// every group. 
void Graph::countGroups() 
{ 
    // Mark all the vertices as not visited 
    bool* visited = new bool[V]; 
    memset(visited, 0, V*sizeof(int)); 
  
    int existing_groups = 0, new_groups = 1; 
    for (int i = 0; i < V; i++) 
    { 
        // If not in any group. 
        if (visited[i] == false)  
        { 
            existing_groups++; 
              
            // Number of new groups that 
            // can be formed. 
            new_groups = new_groups *  
                     countUtil(i, visited); 
        } 
    } 
      
    if (existing_groups == 1) 
        new_groups = 0; 
      
    cout << "No. of existing groups are " 
         << existing_groups << endl; 
    cout << "No. of new groups that can be"
            " formed are " << new_groups  
         << endl; 
} 
  
// Driver code 
int main() 
{ 
    int n = 6; 
  
    // Create a graph given in the above diagram 
    Graph g(n); // total 6 people 
    g.addRelation(1, 2); // 1 and 2 are friends 
    g.addRelation(3, 4); // 3 and 4 are friends 
    g.addRelation(5, 6); // 5 and 6 are friends 
  
    g.countGroups(); 
  
    return 0; 
} 

Output:

No. of existing groups are 3
No. of new groups that can be formed are 8

Time complexity: O(N + R) where N is the the number of people and R is the number of relations.

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