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.
// 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;
} |
// Java program to illustrate Compute the // parity of a number using XOR import 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 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) |
// C# program to illustrate Compute the // parity of a number using XOR using 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 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 ?> |
//JavaScript program to illustrate Compute the // parity of a number using XOR // Generating the look-up table while // pre-processing function 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 table var table = new Array(256).fill(0);
LOOK_UP(table); // Function to find the parity function 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 code var num = 1742346774;
//Function call var result = Parity(num);
// Result is 1 for odd parity, // 0 for even parity console.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)