Count of pairs having bit size at most X and Bitwise OR equal to X
Given a number X, calculate number of possible pairs (a, b) such that bitwise or of a and b is equal to X and number of bits in both a and b is less than equal to number of bits in X.
Input: X = 6
The possible pairs of (a, b) are (4, 6), (6, 4), (6, 6), (6, 2), (4, 2), (6, 0), (2, 6), (2, 4), (0, 6).
Input: X = 21
In total there are 27 pairs possible.
Approach: To solve the problem mentioned above follow the steps given below:
- Iterate through every bit of given number X.
- If the bit is 1 then from the truth table of Bitwise OR we know that there are 3 combinations possible for that given bit in number a and b that is (0, 1), (1, 0), (1, 1) that is 3 possible ways.
- If the bit is 0 then from the truth table of Bitwise OR we know that there is only 1 combination possible for that given bit in number a and b that is (0, 0).
- So our answer will be answer will be 3 ^ (number of on bits in X).
Below is the implementation of above approach:
Time complexity: O(log(X))
Auxiliary Space: O(1)