Finding the Parity of a number Efficiently

Given an integer N. The task is to write a program to find the parity of the given number. 
Note: Parity of a number is used to define if the total number of set-bits(1-bit in binary representation) in a number is even or odd. If the total number of set-bits in the binary representation of a number is even then the number is said to have even parity, otherwise, it will have odd parity.

Examples: 

Input : N = 13 
Output : Odd Parity
Explanation:
Binary representation of 13 is (1101)

Input : N = 9 (1001)
Output : Even Parity

Method 1

The parity of a number represented by 32-bits can be efficiently calculated by performing the following operations.
Let the given number be x, then perform the below operations: 

• y = x^(x>>1)
• y = y^(y>>2)
• y = y^(y>>4)
• y = y^(y>>8)
• y = y^(y>>16)

Now, the rightmost bit in y will represent the parity of x. If the rightmost bit is 1, then x will have odd parity and if it is 0 then x will have even parity.
So, in order to extract the last bit of y, perform bit-wise AND operation of y with 1. 

Why does this work?

Consider that we want to find the parity of n = 150 = 1001 0110 (in binary).

1. Let’s divide this number into two parts and xor them and assign it to n: n = n ^ (n >> 4) = 1001 ^ 0110 = 1111.

Dissimilar bits result in a 1 bit in the result while similar bits result in a 0. We have basically considered all 8 bits to arrive at this intermediate result. So, effectively we have nullified even parities within the number.

Now repeat step 1 again until you end up with a single bit.
n = n ^ (n >> 2) = 11 ^ 11 = 00
n = n ^ (n >> 1) = 0 ^ 0 = 0
Final result = n & 1 = 0

Another example:

n = 1000 0101
n = n ^ (n >> 4) = 1000 ^ 0101 = 1101
n = n ^ (n >> 2) = 11 ^ 01 = 10
n = n ^ (n >> 1) = 1 ^ 0 = 1
Final result = n & 1 = 1

if(y&1==1)
    odd Parity
else
    even Parity

Below is the implementation of the above approach: 

C++

// Program to find the parity of a given number
#include <bits/stdc++.h>

using namespace std;

// Function to find the parity
bool findParity(int x)
{
    int y = x ^ (x >> 1);
    y = y ^ (y >> 2);
    y = y ^ (y >> 4);
    y = y ^ (y >> 8);
    y = y ^ (y >> 16);

    // Rightmost bit of y holds the parity value
    // if (y&1) is 1 then parity is odd else even
    if (y & 1)
        return 1;
    return 0;
}

// Driver code
int main()
{
    (findParity(9)==0)?cout<<"Even Parity\n":
                            cout<<"Odd Parity\n";

    (findParity(13)==0)?cout<<"Even Parity\n":
                            cout<<"Odd Parity\n";

    return 0;
}

Java

// Program to find the
// parity of a given number
import java.io.*;

class GFG
{

// Function to find the parity
static boolean findParity(int x)
{
    int y = x ^ (x >> 1);
        y = y ^ (y >> 2);
        y = y ^ (y >> 4);
        y = y ^ (y >> 8);
        y = y ^ (y >> 16);

    // Rightmost bit of y holds
    // the parity value
    // if (y&1) is 1 then parity
    // is odd else even
    if ((y & 1) > 0)
        return true;
    return false;
}

// Driver code
public static void main (String[] args)
{
    if((findParity(9) == false))
        System.out.println("Even Parity");
    else
        System.out.println("Odd Parity");

    if(findParity(13) == false)
        System.out.println("Even Parity");
    else
        System.out.println("Odd Parity");
}
}

// This Code is Contributed by chandan_jnu.

Python3

# Program to find the
# parity of a given number

# Function to find the parity
def findParity(x):
    y = x ^ (x >> 1);
    y = y ^ (y >> 2);
    y = y ^ (y >> 4);
    y = y ^ (y >> 8);
    y = y ^ (y >> 16);

    # Rightmost bit of y holds
    # the parity value if (y&1)
    # is 1 then parity is odd
    # else even
    if (y & 1):
        return 1;
    return 0;

# Driver code
if(findParity(9) == 0):
    print("Even Parity");
else:
    print("Odd Parity\n");

if(findParity(13) == 0):
    print("Even Parity");
else:
    print("Odd Parity");

# This code is contributed by mits

C#

// Program to find the
// parity of a given number
using System;

class GFG
{

// Function to find the parity
static bool findParity(int x)
{
    int y = x ^ (x >> 1);
        y = y ^ (y >> 2);
        y = y ^ (y >> 4);
        y = y ^ (y >> 8);
        y = y ^ (y >> 16);

    // Rightmost bit of y holds
    // the parity value
    // if (y&1) is 1 then parity
    // is odd else even
    if ((y & 1) > 0)
        return true;
    return false;
}

// Driver code
public static void Main ()
{
    if((findParity(9) == false))
        Console.WriteLine("Even Parity");
    else
        Console.WriteLine("Odd Parity");

    if(findParity(13) == false)
        Console.WriteLine("Even Parity");
    else
        Console.WriteLine("Odd Parity");
}
}

// This Code is Contributed
// by chandan_jnu

PHP

<?php
// Program to find the
// parity of a given number

// Function to find the parity
function findParity($x)
{
    $y = $x ^ ($x >> 1);
    $y = $y ^ ($y >> 2);
    $y = $y ^ ($y >> 4);
    $y = $y ^ ($y >> 8);
    $y = $y ^ ($y >> 16);

    // Rightmost bit of y holds
    // the parity value if (y&1)
    // is 1 then parity is odd
    // else even
    if ($y & 1)
        return 1;
    return 0;
}

// Driver code
(findParity(9) == 0) ?
 print("Even Parity\n"):
 print("Odd Parity\n");

(findParity(13) == 0) ?
print("Even Parity\n"):
print("Odd Parity\n");

// This Code is Contributed by mits
?>

Javascript

<script>
// Javascript Program to find the parity of a given number

// Function to find the parity
function findParity(x) {
    let y = x ^ (x >> 1);
    y = y ^ (y >> 2);
    y = y ^ (y >> 4);
    y = y ^ (y >> 8);
    y = y ^ (y >> 16);

    // Rightmost bit of y holds the parity value
    // if (y&1) is 1 then parity is odd else even
    if (y & 1)
        return 1;
    return 0;
}

// Driver code

(findParity(9) == 0) ? document.write("Even Parity<br>") :
    document.write("Odd Parity<br>");

(findParity(13) == 0) ? document.write("Even Parity<br>") :
    document.write("Odd Parity<br>");

// This code is contributed by _saurabh_jaiswal
</script>

Output

Even Parity
Odd Parity

Time Complexity: O(1)
Auxiliary Space: O(1)   

Method 2

We know that a number with odd parity has an odd number of set bits (1 in binary) in its binary representation, and an even parity number has an even number of 1’s in its binary representation. 

Hence we can simply count the number of 1’s by the following code:

C++

#include <bits/stdc++.h>
using namespace std;

int main()
{
    int number = 7;     //7 means 111 in binary; 3 bits set = odd parity
    bool oddParity = false;

    while(number)   //while number != 0
    {
        //invert the parity because the next statement eliminates one 1
        oddParity = !oddParity;

          //eliminate one 1 from number
          number &= (number-1);
    }

    cout<<oddParity<<endl;

    return 0;
}

Output

1

Time Complexity: O(1)
Auxiliary Space: O(1)   

Note: The number of times the above while loop runs is equal to the number of set bits in the number.

References:

1. Bit Twiddling Hacks from a Stanford University professor