# Compute the parity of a number using XOR and table look-up

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 implementation that **works for both 32 bit and 64 bit** numbers.

## C++

`// 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; ` `} ` |

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## Python3

# Python3 program to illustrate Compute the

# parity of a number using XOR

# Generating the look-up table while

# pre-processing

def P2(n, table):

table.extend([n, n ^ 1, n ^ 1, n])

def P4(n, table):

return (P2(n, table), P2(n ^ 1, table),

P2(n ^ 1, table), P2(n, table))

def P6(n, table):

return (P4(n, table), P4(n ^ 1, table),

P4(n ^ 1, table), P4(n, table))

def LOOK_UP(table):

return (P6(0, table), P6(1, table),

P6(1, table), P6(0, table))

# LOOK_UP is the macro expansion to

# generate the table

table = [0] * 256

LOOK_UP(table)

# Function to find the parity

def Parity(num) :

# Number is considered to be

# of 32 bits

max = 16

# Dividing the number o 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

if __name__ ==”__main__”:

num = 1742346774

# Result is 1 for odd parity,

# 0 for even parity

result = Parity(num)

print(“Odd Parity”) if result else print(“Even Parity”)

# This code is contributed by

# Shubham Singh(SHUBHAMSINGH10)

## PHP

`<?php ` `// PHP program to illustrate ` `// Compute the parity of a ` `// number using XOR ` ` ` `/* 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 ` `$table = array(LOOK_UP ); ` `*/` ` ` `// Function to find ` `// the parity ` `function` `Parity(` `$num` `) ` `{ ` ` ` `global` `$table` `; ` ` ` ` ` `// Number is considered ` ` ` `// to be of 32 bits ` ` ` `$max` `= 16; ` ` ` ` ` `// Dividing the number ` ` ` `// into 8-bit chunks ` ` ` `// while performing X-OR ` ` ` `while` `(` `$max` `>= 8) ` ` ` `{ ` ` ` `$num` `= ` `$num` `^ (` `$num` `>> ` `$max` `); ` ` ` `$max` `= (int)` `$max` `/ 2; ` ` ` `} ` ` ` ` ` `// Masking the number with ` ` ` `// 0xff (11111111) to produce ` ` ` `// valid 8-bit result ` ` ` `return` `$table` `[` `$num` `& 0xff]; ` `} ` ` ` `// Driver code ` `$num` `= 1742346774; ` ` ` `// Result is 1 for odd ` `// parity, 0 for even parity ` `$result` `= Parity(` `$num` `); ` ` ` `// Printing the desired result ` `if` `(` `$result` `== true) ` ` ` `echo` `"Odd Parity"` `; ` ` ` `else` ` ` `echo` `"Even Parity"` `; ` ` ` `// This code is contributed by ajit ` `?> ` |

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**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.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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