# 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;` `}` |

## 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` `?>` |

**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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