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Minimum One Bit Operations to Make Integers Zero

Difficulty: Hard

Problem Description

Given an integer n, you must transform it into 0 using the following operations any number of times:

  • Change the rightmost (0th) bit in the binary representation of n.
  • Change the ith bit in the binary representation of n if the (i-1)th bit is set to 1 and the (i-2)th through 0th bits are set to 0.

Return the minimum number of operations to transform n into 0.

Key Insights

  • The two operations allow you to manipulate the binary representation of the number in a specific way.
  • The first operation always targets the least significant bit (LSB).
  • The second operation can allow you to clear bits further up the binary representation, but it has specific conditions based on adjacent bits.
  • By breaking down the binary representation and counting the necessary operations to clear each bit, we can derive a solution.

Space and Time Complexity

Time Complexity: O(log n)
Space Complexity: O(1)

Solution

The solution involves counting the number of bits set to 1 in the binary representation of the number n, as each bit set to 1 requires at least one operation to clear it. The second operation can sometimes allow for multiple bits to be cleared in a single operation, based on their configuration. The approach can be implemented efficiently by using bit manipulation techniques to check and clear bits.

Code Solutions

def minimum_operations(n: int) -> int:
    operations = 0
    while n > 0:
        operations += 1  # Each operation to clear a bit
        n = n & (n - 1)  # Clear the rightmost set bit
    return operations

# Example usage
print(minimum_operations(3))  # Output: 2
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