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
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.
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.
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.
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.
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).
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