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 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 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
// 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
# 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#
// 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 // 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
//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)
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