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
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 :
3. Split the 32-bit binary representation into
16-bit chunks :
0110011111011010 | 0001101000010110
4. Compute X-OR :
5. Split the 16-bit binary representation
into 8-bit chunks : 01111101 | 11001100
6. Again, Compute X-OR :
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
// 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
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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