Given a positive integer N denoting the number of astronauts(labelled from 0 from (N – 1))and a matrix mat[][] containing the pairs of astronauts that are from the same country, the task is to count the number of ways to choose two astronauts from different countries.
Examples:
Input: N = 6, mat[][] = {{0, 1}, {0, 2}, {2, 5}}
Output: 9
Explanation:
Astronauts with ID {0, 1, 2, 5} belong to first country, astronaut with ID {3} belongs to second country and astronaut with ID {4} belongs to third country. The number of ways to choose two astronauts from different countries is:
- Choose 1 astronaut from country 1 and 1 astronaut from country 2, then the total number of ways is 4*1 = 4.
- Choose 1 astronaut from country 1 and 1 astronaut from country 3, then the total number of ways is 4*1 = 4.
- Choose 1 astronaut from country 2 and 1 astronaut from country 3, then the total number of ways is 1*1 = 1.
Therefore, the total number of ways is 4 + 4 + 1 = 9.
Input: N = 5, mat[][] = {{0, 1}, {2, 3}, {0, 4}}
Output: 6
Approach: The given problem can be solved by modeling this problem as a graph in which astronauts represent the vertices of the graph and the given pairs represent the edges in the graph. After constructing the graph, the idea is to calculate the number of ways to select 2 astronauts from different countries. Follow the steps to solve the problem:
- Create a list of lists, adj[][] to store the adjacency list of the graph.
- Traverse the given array arr[] using the variable i and append arr[i][1] to adj[arr[i][0]] and also append arr[i][0] to adj[arr[i][1]] for the undirected edge.
- Now find the size of each connected component of the graph by performing the Depth First Search, using the approach discussed in this article, and store all the sizes of connected components be stored in an array say v[].
- Initialize an integer variable, say ans as the total number of ways to choose 2 astronauts from N astronauts i.e., N*(N – 1)/2.
- Traverse the array v[] and subtract v[i]*(v[i] – 1)/2 from the variable ans to exclude all possible pairs among each connected components.
- After completing the above steps, print the value of ans as the result.
Below is the implementation of the above approach:
C++
// C++ program for the above approach#include <bits/stdc++.h>using namespace std;// Function to perform the DFS Traversal// to find the count of connected// componentsvoid dfs(int v, vector<vector<int> >& adj, vector<bool>& visited, int& num){ // Marking vertex visited visited[v] = true; num++; // DFS call to neighbour vertices for (int i = 0; i < adj[v].size(); i++) { // If the current node is not // visited, then recursively // call DFS if (!visited[adj[v][i]]) { dfs(adj[v][i], adj, visited, num); } }}// Function to find the number of ways// to choose two astronauts from the// different countriesvoid numberOfPairs( int N, vector<vector<int> > arr){ // Stores the Adjacency list vector<vector<int> > adj(N); // Constructing the graph for (vector<int>& i : arr) { adj[i[0]].push_back(i[1]); adj[i[1]].push_back(i[0]); } // Stores the visited vertices vector<bool> visited(N); // Stores the size of every // connected components vector<int> v; int num = 0; for (int i = 0; i < N; i++) { if (!visited[i]) { // DFS call to the graph dfs(i, adj, visited, num); // Store size of every // connected component v.push_back(num); num = 0; } } // Stores the total number of // ways to count the pairs int ans = N * (N - 1) / 2; // Traverse the array for (int i : v) { ans -= (i * (i - 1) / 2); } // Print the value of ans cout << ans;}// Driver Codeint main(){ int N = 6; vector<vector<int> > arr = { { 0, 1 }, { 0, 2 }, { 2, 5 } }; numberOfPairs(N, arr); return 0;} |
Java
// Java program for the above approachimport java.util.*;public class GFG{ // Function to perform the DFS Traversal // to find the count of connected // components static Vector<Vector<Integer>> adj; static boolean[] visited; static int num; // Function to perform the DFS Traversal // to find the count of connected // components static void dfs(int v) { // Marking vertex visited visited[v] = true; num++; // DFS call to neighbour vertices for (int i = 0; i < adj.get(v).size(); i++) { // If the current node is not // visited, then recursively // call DFS if (!visited[adj.get(v).get(i)]) { dfs(adj.get(v).get(i)); } } } // Function to find the number of ways // to choose two astronauts from the // different countries static void numberOfPairs(int N, int[][] arr) { // Stores the Adjacency list adj = new Vector<Vector<Integer>>(); for(int i = 0; i < N; i++) { adj.add(new Vector<Integer>()); } // Constructing the graph for (int i = 0; i < 2; i++) { adj.get(arr[i][0]).add(arr[i][1]); adj.get(arr[i][1]).add(arr[i][0]); } // Stores the visited vertices visited = new boolean[N]; Arrays.fill(visited, false); // Stores the size of every // connected components Vector<Integer> v = new Vector<Integer>(); num = 0; for (int i = 0; i < N; i++) { if (!visited[i]) { // DFS call to the graph dfs(i); // Store size of every // connected component v.add(num); num = 0; } } // Stores the total number of // ways to count the pairs int ans = N * (N - 1) / 2 + 1; // Traverse the array for (int i = 0; i < v.size(); i++) { ans -= (v.get(i) * (v.get(i) - 1) / 2) +1; } // Print the value of ans System.out.print(ans); } public static void main(String[] args) { int N = 6; int[][] arr = { { 0, 1 }, { 0, 2 }, { 2, 5 } }; numberOfPairs(N, arr); }}// This code is contributed by suresh07 |
Python3
# Python3 program for the above approach # Function to perform the DFS Traversal# to find the count of connected# componentsadj = []visited = []num = 0 def dfs(v): global adj, visited, num # Marking vertex visited visited[v] = True num+=1 # DFS call to neighbour vertices for i in range(len(adj[v])): # If the current node is not # visited, then recursively # call DFS if (not visited[adj[v][i]]): dfs(adj[v][i])# Function to find the number of ways# to choose two astronauts from the# different countriesdef numberOfPairs(N, arr): global adj, visited, num # Stores the Adjacency list adj = [] for i in range(N): adj.append([]) # Constructing the graph for i in range(len(arr)): adj[arr[i][0]].append(arr[i][1]) adj[arr[i][1]].append(arr[i][0]) # Stores the visited vertices visited = [False]*(N) # Stores the size of every # connected components v = [] num = 0 for i in range(N): if (not visited[i]): # DFS call to the graph dfs(i) # Store size of every # connected component v.append(num) num = 0 # Stores the total number of # ways to count the pairs ans = N * int((N - 1) / 2) # Traverse the array for i in range(len(v)): ans -= (v[i] * int((v[i] - 1) / 2)) ans+=1 # Print the value of ans print(ans)N = 6arr = [ [ 0, 1 ], [ 0, 2 ], [ 2, 5 ] ]numberOfPairs(N, arr)# This code is contributed by mukesh07. |
C#
// C# program for the above approachusing System;using System.Collections.Generic;class GFG { // Function to perform the DFS Traversal // to find the count of connected // components static List<List<int>> adj; static bool[] visited; static int num; // Function to perform the DFS Traversal // to find the count of connected // components static void dfs(int v) { // Marking vertex visited visited[v] = true; num++; // DFS call to neighbour vertices for (int i = 0; i < adj[v].Count; i++) { // If the current node is not // visited, then recursively // call DFS if (!visited[adj[v][i]]) { dfs(adj[v][i]); } } } // Function to find the number of ways // to choose two astronauts from the // different countries static void numberOfPairs(int N, int[,] arr) { // Stores the Adjacency list adj = new List<List<int>>(); for(int i = 0; i < N; i++) { adj.Add(new List<int>()); } // Constructing the graph for (int i = 0; i < 2; i++) { adj[arr[i,0]].Add(arr[i,1]); adj[arr[i,1]].Add(arr[i,0]); } // Stores the visited vertices visited = new bool[N]; Array.Fill(visited, false); // Stores the size of every // connected components List<int> v = new List<int>(); num = 0; for (int i = 0; i < N; i++) { if (!visited[i]) { // DFS call to the graph dfs(i); // Store size of every // connected component v.Add(num); num = 0; } } // Stores the total number of // ways to count the pairs int ans = N * (N - 1) / 2 + 1; // Traverse the array for (int i = 0; i < v.Count; i++) { ans -= (v[i] * (v[i] - 1) / 2) +1; } // Print the value of ans Console.Write(ans); } static void Main() { int N = 6; int[,] arr = { { 0, 1 }, { 0, 2 }, { 2, 5 } }; numberOfPairs(N, arr); }}// This code is contributed by divyesh072019. |
Javascript
<script> // Javascript program for the above approach // Function to perform the DFS Traversal // to find the count of connected // components let adj; let visited; let num; function dfs(v) { // Marking vertex visited visited[v] = true; num++; // DFS call to neighbour vertices for (let i = 0; i < adj[v].length; i++) { // If the current node is not // visited, then recursively // call DFS if (!visited[adj[v][i]]) { dfs(adj[v][i]); } } } // Function to find the number of ways // to choose two astronauts from the // different countries function numberOfPairs(N, arr) { // Stores the Adjacency list adj = []; for(let i = 0; i < N; i++) { adj.push([]); } // Constructing the graph for (let i = 0; i < arr.length; i++) { adj[arr[i][0]].push(arr[i][1]); adj[arr[i][1]].push(arr[i][0]); } // Stores the visited vertices visited = new Array(N); visited.fill(false); // Stores the size of every // connected components let v = []; num = 0; for (let i = 0; i < N; i++) { if (!visited[i]) { // DFS call to the graph dfs(i); // Store size of every // connected component v.push(num); num = 0; } } // Stores the total number of // ways to count the pairs let ans = N * (N - 1) / 2; // Traverse the array for (let i = 0; i < v.length; i++) { ans -= (v[i] * (v[i] - 1) / 2); } // Print the value of ans document.write(ans); } let N = 6; let arr = [ [ 0, 1 ], [ 0, 2 ], [ 2, 5 ] ]; numberOfPairs(N, arr);// This code is contributed by rameshtravel07.</script> |
Output:
9
Time Complexity: O(N + E), where N is the number of vertices and E is the number of edges.
Auxiliary Space: O(N + E)
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