Parity of a number refers to whether it contains an odd or even number of 1-bits. The number has “odd parity”, if it contains odd number of 1-bits and is “even parity” if it contains even number of 1-bits.

1 --> parity of the set is odd 0 --> parity of the set is even

**Examples:**

Input : 254 Output : Odd ParityExplanation :Binary of 254 is 11111110. There are 7 ones. Thus, parity is odd. Input : 1742346774 Output : Even

**Method 1 : (Naive approach)**

We have already discussed this method here.

**Method 2 : (Efficient)**

Pr-requisites : Table look up, X-OR magic

If we break a number S into two parts S_{1} and S_{2} such **S = S _{1}S_{2}**. If we know parity of S

_{1}and S

_{2}, we can compute parity of S using below facts :

- If S
_{1}and S_{2}have the same parity, i.e. they both have an even number of bits or an odd number of bits, their union S will have an even number of bits. - Therefore parity of S is XOR of parities of S
_{1}and S_{2}

The idea is to create a look up table to store parities of all 8 bit numbers. Then compute parity of whole number by dividing it into 8 bit numbers and using above facts.

**Steps:**

1. Create a look-up table for 8-bit numbers ( 0 to 255 ) Parity of 0 is 0. Parity of 1 is 1. . . . Parity of 255 is 0. 2. Break the number into 8-bit chunks while performing XOR operations. 3. Check for the result in the table for the 8-bit number.

Since a 32 bit or 64 bit number contains constant number of bytes, the above steps take O(1) time.

Example :

1. Take 32-bit number :17423467742. Calculate Binary of the number :011001111101101000011010000101103. Split the 32-bit binary representation into 16-bit chunks :0110011111011010 | 00011010000101104. Compute X-OR : 0110011111011010 ^ 0001101000010110 ___________________ = 0111110111001100 5. Split the 16-bit binary representation into 8-bit chunks : 01111101 | 11001100 6. Again, Compute X-OR : 01111101 ^ 11001100 ___________________ = 1011000110110001 is 177 in decimal. Check for its parity in look-up table :Even number of 1 = Even parity.Thus, Parity of 1742346774 is even.

Below is the C++ implementation that **works for both 32 bit and 64 bit** numbers.

// CPP program to illustrate Compute the parity of a // number using XOR #include <bits/stdc++.h> // Generating the look-up table while pre-processing #define P2(n) n, n ^ 1, n ^ 1, n #define P4(n) P2(n), P2(n ^ 1), P2(n ^ 1), P2(n) #define P6(n) P4(n), P4(n ^ 1), P4(n ^ 1), P4(n) #define LOOK_UP P6(0), P6(1), P6(1), P6(0) // LOOK_UP is the macro expansion to generate the table unsigned int table[256] = { LOOK_UP }; // Function to find the parity int Parity(int num) { // Number is considered to be of 32 bits int max = 16; // Dividing the number into 8-bit // chunks while performing X-OR while (max >= 8) { num = num ^ (num >> max); max = max / 2; } // Masking the number with 0xff (11111111) // to produce valid 8-bit result return table[num & 0xff]; } // Driver code int main() { unsigned int num = 1742346774; // Result is 1 for odd parity, 0 for even parity bool result = Parity(num); // Printing the desired result result ? std::cout << "Odd Parity" : std::cout << "Even Parity"; return 0; }

Output:

Even Parity

Time Complexity : O(1). Note that a 32 bit or 64 bit number has fixed number of bytes (4 in case of 32 bits and 8 in case of 64 bits).

This article is contributed by **Rohit Thapliyal**. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

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