Count of pairs that can be removed from Array without changing the Mean

Last Updated : 28 Apr, 2022

Given an array arr[] of size N, the task is to find the number of pairs of the array upon removing which the mean (i.e. the average of all elements) of the array remains unchanged.

Examples:

Input: N = 5, arr[] = {1, 4, 7, 3, 5}
Output: 2
Explanation: The required pairs of positions are – {0, 2} and {3, 4}.
Mean of original array = (1 + 4 + 7 + 3 + 5) / 5 = 4.
On deletion of elements at positions 0 and 2, array becomes {4, 3, 5}.
Mean of the new array = (4 + 3 + 5) / 3 = 4. which is same as the mean of original array.
On deletion of elements at positions 3 and 4, array becomes {1, 4, 7}.
Mean of the new array = (1 + 4 + 7) / 3 = 4. which is same as the mean of original array.

Input: N = 3, A = {50, 20, 10}
Output: 0
Explanation: No such pair is possible.

Approach: Consider the below observation:

Observation:

If a sum (S = 2*mean) is subtracted from the original array (with initial mean = M), the mean of the new updated array will still remains as M.

Derivation:

Sum of array (of size N) = N * mean = N * M

new mean = (sum of array – 2 * M) / (N – 2)
=(N*M – 2*M)/(N – 2)
= M.

The following steps can be followed to solve the problem –

Calculate mean of the given array.
Calculate required sum S i.e. 2*mean.
If S is not an integer, return 0, since there will not be any pair with a nonintegral sum.
Else, find a number of required pairs by hashing.

Below is the implementation of the above approach:

C++

// C++ code to implement the above approach
#include <bits/stdc++.h>
using namespace std;
 
// Function to calculate the number
// of pairs of positions [i, j] (i<j),
// such that on deletion of elements
// at these positions, the mean of
// remaining (n-2) elements does not change
int countPairs(int n, int A[])
{
    // Calculating sum of array
    int sum = 0;
    for (int i = 0; i < n; i++) {
        sum += A[i];
    }
 
    // Calculating mean of array
    double mean = (sum * 1.0) / n;
 
    // Required sum = twice of mean
    double required_sum = 2.0 * mean;
 
    // Checking if required_sum is integral
    int check = required_sum;
    double temp = required_sum - check;
 
    if (temp > 0) {
        return 0;
    }
    else {
 
        // Initialising count variable to
        // store total number of valid pairs
        int count = 0;
 
        // Declaring a map to calculate
        // number of pairs with required sum
        map<int, int> mp;
 
        // Standard procedure to calculate
        // total number of pairs
        // of required sum
        for (int i = 0; i < n; i++) {
            if (i > 0) {
                count += mp[required_sum
                            - A[i]];
            }
            mp[A[i]]++;
        }
 
        // Returning count
        return count;
    }
 
    // If no pairs with required sum
    return 0;
}
 
// Driver code
int main()
{
    int N = 5;
    int arr[] = { 1, 4, 7, 3, 5 };
    int numberOfPairs = countPairs(N, arr);
    cout << numberOfPairs;
    return 0;
}

Java

// Java code to implement the approach
import java.io.*;
import java.util.*;
 
class GFG
{
 
  // Function to calculate the number
  // of pairs of positions [i, j] (i<j),
  // such that on deletion of elements
  // at these positions, the mean of
  // remaining (n-2) elements does not change
  public static int countPairs(int n, int A[])
  {
 
    // Calculating sum of array
    int sum = 0;
    for (int i = 0; i < n; i++) {
      sum += A[i];
    }
 
    // Calculating mean of array
    double mean = (sum * 1.0) / n;
 
    // Required sum = twice of mean
    double required_sum = 2.0 * mean;
 
    // Checking if required_sum is integral
    int check = (int)required_sum;
    double temp = required_sum - check;
 
    if (temp > 0) {
      return 0;
    }
    else {
 
      // Initialising count variable to
      // store total number of valid pairs
      int count = 0;
 
      // Declaring a map to calculate
      // number of pairs with required sum
      TreeMap<Integer, Integer> mp
        = new TreeMap<Integer, Integer>();
 
      // Standard procedure to calculate
      // total number of pairs
      // of required sum
      for (int i = 0; i < n; i++) {
        if (i > 0) {
          if (mp.get((int)required_sum - A[i]) != null)
          {
            count += mp.get((int)required_sum - A[i]);
          }
 
        }
        if (mp.get(A[i]) != null)
        {
          mp.put(A[i],mp.get(A[i])+1);
        }
        else
        {
          mp.put(A[i],1);
        }
      }
 
      // Returning count
      return count;
    }
 
  }
 
  public static void main(String[] args)
  {
    int N = 5;
    int arr[] = { 1, 4, 7, 3, 5 };
    int numberOfPairs = countPairs(N, arr);
    System.out.print(numberOfPairs);
  }
}
 
// This code is contributed by Pushpesh Raj

Python

# Python code to implement the above approach
 
# Function to calculate the number
# of pairs of positions [i, j] (i<j),
# such that on deletion of elements
# at these positions, the mean of
# remaining (n-2) elements does not change
def countPairs(n, A):
     
    # Calculating sum of array
    sum = 0
    for i in range(0, n):
        sum += A[i]
 
    # Calculating mean of array
    mean = (sum * 1.0) / n
 
    # Required sum = twice of mean
    required_sum = 2.0 * mean
 
    # Checking if required_sum is integral
    check = required_sum
    temp = required_sum - check
 
    if (temp > 0):
        return 0
         
    else:
 
        # Initialising count variable to
        # store total number of valid pairs
        count = 0
 
        # Declaring a map to calculate
        # number of pairs with required sum
        mp = {}
 
        # Standard procedure to calculate
        # total number of pairs
        # of required sum
        for i in range(0, n):
            if (i > 0 and required_sum - A[i] in mp):
                count += mp[required_sum - A[i]]
             
            if arr[i] in mp:
                mp[arr[i]] += 1
            else:
                mp[arr[i]] = 1
 
        # Returning count
        return count
 
    # If no pairs with required sum
    return 0
 
# Driver code
 
N = 5
arr = [ 1, 4, 7, 3, 5 ]
numberOfPairs = countPairs(N, arr)
print(numberOfPairs)
 
# This code is contributed by Samim Hossain Mondal.

C#

// C# program to implement
// the above approach
using System;
using System.Collections.Generic;
 
class GFG
{
 
  // Function to calculate the number
  // of pairs of positions [i, j] (i<j),
  // such that on deletion of elements
  // at these positions, the mean of
  // remaining (n-2) elements does not change
  static int countPairs(int n, int[] A)
  {
    // Calculating sum of array
    int sum = 0;
    for (int i = 0; i < n; i++) {
      sum += A[i];
    }
 
    // Calculating mean of array
    double mean = (sum * 1.0) / n;
 
    // Required sum = twice of mean
    double required_sum = 2.0 * mean;
 
    // Checking if required_sum is integral
    int check = (int)required_sum;
    double temp = required_sum - check;
 
    if (temp > 0) {
      return 0;
    }
    else {
 
      // Initialising count variable to
      // store total number of valid pairs
      int count = 0;
 
      // Declaring a map to calculate
      // number of pairs with required sum
      Dictionary<int,
      int> mp = new Dictionary<int,
      int>();
 
      // Standard procedure to calculate
      // total number of pairs
      // of required sum
      for (int i = 0; i < n; i++) {
        if (i > 0 && ( mp.ContainsKey((int)required_sum - A[i]))) {
          count += mp[(int)required_sum
                      - A[i]];
        }
 
        if (mp.ContainsKey(A[i]))
        {
          int x = mp[A[i]];
          mp[A[i]]= x + 1;
        }
        else
          mp.Add(A[i], 1);
      }
 
      // Returning count
      return count;
    }
 
    // If no pairs with required sum
    return 0;
  }
 
  // Driver Code
  public static void Main()
  {
    int N = 5;
    int[] arr = { 1, 4, 7, 3, 5 };
    int numberOfPairs = countPairs(N, arr);
    Console.Write(numberOfPairs);
  }
}
 
// This code is contributed by sanjoy_62.

Javascript

<script>
    // JavaScript code to implement the above approach
 
    // Function to calculate the number
    // of pairs of positions [i, j] (i<j),
    // such that on deletion of elements
    // at these positions, the mean of
    // remaining (n-2) elements does not change
    const countPairs = (n, A) => {
        // Calculating sum of array
        let sum = 0;
        for (let i = 0; i < n; i++) {
            sum += A[i];
        }
 
        // Calculating mean of array
        let mean = (sum * 1.0) / n;
 
        // Required sum = twice of mean
        let required_sum = 2.0 * mean;
 
        // Checking if required_sum is integral
        let check = required_sum;
        let temp = required_sum - check;
 
        if (temp > 0) {
            return 0;
        }
        else {
 
            // Initialising count variable to
            // store total number of valid pairs
            let count = 0;
 
            // Declaring a map to calculate
            // number of pairs with required sum
            let mp = {};
 
            // Standard procedure to calculate
            // total number of pairs
            // of required sum
            for (let i = 0; i < n; i++) {
                if (i > 0 && (required_sum - A[i] in mp)) {
                    count += mp[required_sum
                        - A[i]];
                }
                if (A[i] in mp) mp[A[i]] += 1;
                else mp[A[i]] = 1;
            }
 
            // Returning count
            return count;
        }
 
        // If no pairs with required sum
        return 0;
    }
 
    // Driver code
 
    let N = 5;
    let arr = [1, 4, 7, 3, 5];
    let numberOfPairs = countPairs(N, arr);
    document.write(numberOfPairs);
 
    // This code is contributed by rakeshsahni
 
</script>

Output

Time Complexity: O(N)
Auxiliary Space: O(N)

Count pairs with odd Bitwise XOR that can be removed and replaced by their Bitwise OR

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Count of pairs that can be removed from Array without changing the Mean

C++

Java

Python

C#

Javascript

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