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 Parity Explanation : 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) Pre-requisites : Table look up, X-OR magic If we break a number S into two parts S1 and S2 such S = S1S2. If we know parity of S1 and S2, we can compute parity of S using below facts :
- If S1 and S2 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 S1 and S2
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 : 1742346774 2. Calculate Binary of the number : 01100111110110100001101000010110 3. Split the 32-bit binary representation into 16-bit chunks : 0110011111011010 | 0001101000010110 4. 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 ___________________ = 10110001 10110001 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 tableunsigned int table[256] = { LOOK_UP };// Function to find the parityint 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 codeint 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;} |
Java
// Java program to illustrate Compute the// parity of a number using XORimport java.util.ArrayList;class GFG { // LOOK_UP is the macro expansion to // generate the table static ArrayList<Integer> table = new ArrayList<Integer>(); // Generating the look-up table while // pre-processing static void P2(int n) { table.add(n); table.add(n ^ 1); table.add(n ^ 1); table.add(n); } static void P4(int n) { P2(n); P2(n ^ 1); P2(n ^ 1); P2(n); } static void P6(int n) { P4(n); P4(n ^ 1); P4(n ^ 1); P4(n) ; } static void LOOK_UP() { P6(0); P6(1); P6(1); P6(0); } // Function to find the parity static int Parity(int num) { // Number is considered to be // of 32 bits int 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.get(num & 0xff); } public static void main(String[] args) { // Driver code int num = 1742346774; LOOK_UP(); //Function call int result = Parity(num); // Result is 1 for odd parity, // 0 for even parity if (result != 0) System.out.println("Odd Parity"); else System.out.println("Even Parity"); }}//This code is contributed by phasing17 |
Python3
# Python3 program to illustrate Compute the# parity of a number using XOR# Generating the look-up table while# pre-processingdef 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 tabletable = [0] * 256LOOK_UP(table)# Function to find the paritydef 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 codeif __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) |
C#
// C# program to illustrate Compute the// parity of a number using XORusing System;using System.Collections.Generic;class GFG { // LOOK_UP is the macro expansion to // generate the table static List<int> table = new List<int>(); // Generating the look-up table while // pre-processing static void P2(int n) { table.Add(n); table.Add(n ^ 1); table.Add(n ^ 1); table.Add(n); } static void P4(int n) { P2(n); P2(n ^ 1); P2(n ^ 1); P2(n); } static void P6(int n) { P4(n); P4(n ^ 1); P4(n ^ 1); P4(n); } static void LOOK_UP() { P6(0); P6(1); P6(1); P6(0); } // Function to find the parity static int Parity(int num) { // Number is considered to be // of 32 bits int 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]; } public static void Main(string[] args) { // Driver code int num = 1742346774; LOOK_UP(); // Function call int result = Parity(num); // Result is 1 for odd parity, // 0 for even parity if (result != 0) Console.WriteLine("Odd Parity"); else Console.WriteLine("Even Parity"); }}// This code is contributed by phasing17 |
PHP
<?php// PHP program to illustrate// Compute the parity of a// number using XOR/* Generating the look-uptable 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 expansionto generate the table$table = array(LOOK_UP );*/// Function to find// the parityfunction Parity($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($num);// Printing the desired resultif($result == true) echo "Odd Parity" ; else echo"Even Parity";// This code is contributed by ajit?> |
Javascript
//JavaScript program to illustrate Compute the// parity of a number using XOR// Generating the look-up table while// pre-processingfunction P2(n, table){ table.push(n, n ^ 1, n ^ 1, n);}function P4(n, table){ return (P2(n, table), P2(n ^ 1, table), P2(n ^ 1, table), P2(n, table));}function P6(n, table){ return (P4(n, table), P4(n ^ 1, table), P4(n ^ 1, table), P4(n, table)) ;}function 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 tablevar table = new Array(256).fill(0);LOOK_UP(table);// Function to find the parityfunction Parity(num){ // Number is considered to be // of 32 bits var max = 16; // Dividing the number o 8-bit // chunks while performing X-OR while (max >= 8) { num = num ^ (num >> max); max = Math.floor(max / 2); } // Masking the number with 0xff (11111111) // to produce valid 8-bit result return table[num & 0xff] ;}// Driver codevar num = 1742346774; //Function callvar result = Parity(num);// Result is 1 for odd parity,// 0 for even parityconsole.log(result ? "Odd Parity" : "Even Parity");// This code is contributed by phasing17 |
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).
Auxiliary Space: O(1)
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