Count of even and odd set bit with array element after XOR with K

Given an array arr[] and a number K. The task is to find the count of the element having odd and even number of the set-bit after taking XOR of K with every element of the given arr[].

Examples:

Input: arr[] = {4, 2, 15, 9, 8, 8}, K = 3
Output: Even = 2, Odd = 4
Explanation:
The binary representation of the element after taking XOR with K are:
4 ^ 3 = 7 (111)
2 ^ 3 = 1 (1)
15 ^ 3 = 12 (1100)
9 ^ 3 = 10 (1010)
8 ^ 3 = 11 (1011)
8 ^ 3 = 11 (1011)
No of elements with even no of 1’s in binary representation : 2 (12, 10)
No of elements with odd no of 1’s in binary representation : 4 (7, 1, 11, 11)

Input: arr[] = {4, 2, 15, 9, 8, 8}, K = 6
Output: Even = 4, Odd = 2

Naive Approach: The naive approach is to take bitwise XOR of K with each element of the given array arr[] and then, count the set-bit for each element in the array after taking XOR with K.



Time Complexity: O(N*K)

Efficient Approach:
With the help of the following observation, we have:

For Example:
If A = 4 and K = 3
Binary Representation:
A = 100
K = 011
A^K = 111
Therefore, the XOR of number A(which has an odd number of set-bit) with the number K(which has an even number of set-bit) results in an odd number of set-bit.

And If A = 4 and K = 7
Binary Representation:
A = 100
K = 111
A^K = 011
Therefore, the XOR of number A(which has an odd number of set-bit) with the number K(which has an odd number of set-bit) results in an even number of set-bit.

From the above observations:

  • If K has an odd number of set-bit, then the count of elements of array arr[] with even set-bit and odd set-bit after taking XOR with K, will be same as the count of even set-bit and odd set-bit int the array before XOR.
  • If K has an even number of set-bit, then the count of elements of array arr[] with even set-bit and odd set-bit after taking XOR with K, will reverse as the count of even set-bit and odd set-bit in the array before XOR.

Steps:

  1. Store the count of elements having even set-bit and odd set-bit from the given array arr[].
  2. If K has odd set-bit, then the count of even and odd set-bit after XOR with K will be the same as the count of even and odd set-bit calculated above.
  3. If K has even set-bit, then the count of even and odd set-bit after XOR with K will be the count of odd and even set-bit calculated above.

Below is the implementation of the above approach:

C++

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// C++ program to count the set
// bits after taking XOR with a
// number K
#include <bits/stdc++.h>
using namespace std;
  
// Function to store EVEN and odd variable
void countEvenOdd(int arr[], int n, int K)
{
    int even = 0, odd = 0;
  
    // Store the count of even and
    // odd set bit
    for (int i = 0; i < n; i++) {
  
        // Count the set bit using
        // in built function
        int x = __builtin_popcount(arr[i]);
        if (x % 2 == 0)
            even++;
        else
            odd++;
    }
  
    int y;
  
    // Count of set-bit of K
    y = __builtin_popcount(K);
  
    // If y is odd then, count of
    // even and odd set bit will
    // be interchanged
    if (y & 1) {
        cout << "Even = " << odd
             << ", Odd = " << even;
    }
  
    // Else it will remain same as
    // the original array
    else {
        cout << "Even = " << even
             << ", Odd = " << odd;
    }
}
  
// Driver's Code
int main(void)
{
    int arr[] = { 4, 2, 15, 9, 8, 8 };
    int K = 3;
    int n = sizeof(arr) / sizeof(arr[0]);
  
    // Function call to count even
    // and odd
    countEvenOdd(arr, n, K);
    return 0;
}

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Java

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// Java program to count the set
// bits after taking XOR with a
// number K
class GFG {
  
      
    /* Function to get no of set  
    bits in binary representation  
    of positive integer n */
    static int __builtin_popcount(int n) 
    
        int count = 0
        while (n > 0) { 
            count += n & 1
            n >>= 1
        
        return count; 
    }
      
    // Function to store EVEN and odd variable
    static void countEvenOdd(int arr[], int n, int K)
    {
        int even = 0, odd = 0;
      
        // Store the count of even and
        // odd set bit
        for (int i = 0; i < n; i++) {
      
            // Count the set bit using
            // in built function
            int x = __builtin_popcount(arr[i]);
            if (x % 2 == 0)
                even++;
            else
                odd++;
        }
      
        int y;
      
        // Count of set-bit of K
        y = __builtin_popcount(K);
      
        // If y is odd then, count of
        // even and odd set bit will
        // be interchanged
        if ((y & 1) != 0) {
            System.out.println("Even = "+ odd + ", Odd = " + even);
        }
      
        // Else it will remain same as
        // the original array
        else {
            System.out.println("Even = " + even + ", Odd = " + odd);
        }
    }
      
    // Driver's Code
    public static void main (String[] args)
    {
        int arr[] = { 4, 2, 15, 9, 8, 8 };
        int K = 3;
        int n = arr.length;
      
        // Function call to count even
        // and odd
        countEvenOdd(arr, n, K);
    }
   
}
// This code is contributed by Yash_R

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Python3

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# Python3 program to count the set 
# bits after taking XOR with a 
# number K 
  
# Function to store EVEN and odd variable 
def countEvenOdd(arr, n, K) : 
  
    even = 0; odd = 0
  
    # Store the count of even and 
    # odd set bit 
    for i in range(n) :
  
        # Count the set bit using 
        # in built function 
        x = bin(arr[i]).count('1'); 
        if (x % 2 == 0) :
            even += 1
        else :
            odd += 1
  
    # Count of set-bit of K 
    y = bin(K).count('1'); 
  
    # If y is odd then, count of 
    # even and odd set bit will 
    # be interchanged 
    if (y & 1) :
        print("Even =",odd ,", Odd =", even); 
  
    # Else it will remain same as 
    # the original array 
    else :
        print("Even =" , even ,", Odd =", odd); 
  
  
# Driver's Code 
if __name__ == "__main__" :
      
    arr = [ 4, 2, 15, 9, 8, 8 ]; 
    K = 3
    n = len(arr); 
  
    # Function call to count even 
    # and odd 
    countEvenOdd(arr, n, K); 
      
# This code is contributed by Yash_R

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C#

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// C# program to count the set
// bits after taking XOR with a
// number K
using System;
  
public class GFG {
  
      
    /* Function to get no of set  
    bits in binary representation  
    of positive integer n */
    static int __builtin_popcount(int n) 
    
        int count = 0; 
        while (n > 0) { 
            count += n & 1; 
            n >>= 1; 
        
        return count; 
    }
      
    // Function to store EVEN and odd variable
    static void countEvenOdd(int []arr, int n, int K)
    {
        int even = 0, odd = 0;
      
        // Store the count of even and
        // odd set bit
        for (int i = 0; i < n; i++) {
      
            // Count the set bit using
            // in built function
            int x = __builtin_popcount(arr[i]);
            if (x % 2 == 0)
                even++;
            else
                odd++;
        }
      
        int y;
      
        // Count of set-bit of K
        y = __builtin_popcount(K);
      
        // If y is odd then, count of
        // even and odd set bit will
        // be interchanged
        if ((y & 1) != 0) {
            Console.WriteLine("Even = "+ odd + ", Odd = " + even);
        }
      
        // Else it will remain same as
        // the original array
        else {
            Console.WriteLine("Even = " + even + ", Odd = " + odd);
        }
    }
      
    // Driver's Code
    public static void Main (string[] args)
    {
        int []arr = { 4, 2, 15, 9, 8, 8 };
        int K = 3;
        int n = arr.Length;
      
        // Function call to count even
        // and odd
        countEvenOdd(arr, n, K);
    }
   
}
// This code is contributed by Yash_R

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Output:

Even = 2, Odd = 4

Time Complexity: O(N)

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Improved By : Yash_R