Problem Link - Binary Shuffle
Problem Statement
Chef has two integers A and B. He can perform the following operation on A an arbitrary number of times (including zero):
- write A as a binary number with an arbitrary number of leading zeroes (possibly without any)
- shuffle the binary digits of A in an arbitrary way, obtaining a number s
- replace A by s+1
Chef is wondering about the minimum number of operations he has to perform on A in order to obtain B. Compute this number or determine that it is impossible.
Approach
The code idea is to determine how many operations are needed to transform A into B based on their binary representations. It first checks if A is equal to B. If they are equal, the answer is 0, meaning no operations are needed.
Next, the code handles special cases:
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If B=0, it’s impossible to reach 0 from any positive A (since the operations will always yield at least 1), so the answer is -1.
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If B=1:
- If A=0, one operation is required to reach 1 by transforming 0 to 1.
- If A \neq 0, it’s impossible to reach 1 from any non-zero A, resulting in an answer of -1.
For other values of B, the code counts the number of set bits (1s) in both A and B−1 (using the __builtin_popcountll function). Let aa be the count of 1s in A, and bb be the count of 1s in B−1.
Time Complexity
The time complexity is O(\log(\text{max}(A, B))) due to the operations on the binary representations of A and B.
Space Complexity
The space complexity is O(1) because the solution uses a constant amount of space for variables.