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.
Examples:
Input: X = 6
Output: 9
Explanation:
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
Output: 27
Explanation:
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:
// C++ implementation to Count number of // possible pairs of (a, b) such that // their Bitwise OR gives the value X #include <iostream> using namespace std; // Function to count the pairs int count_pairs(int x) { // Initializing answer with 1 int ans = 1; // Iterating through bits of x while (x > 0) { // check if bit is 1 if (x % 2 == 1) // multiplying ans by 3 // if bit is 1 ans = ans * 3; x = x / 2; } return ans; } // Driver code int main() { int X = 6; cout << count_pairs(X) << endl; return 0; } |
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Output:
9
Time complexity: O(log(X))
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