Minimum Operations to Reduce an Integer to 0

Problem Description

Given a positive integer n, reduce it to 0 using the minimum number of operations. In each operation, you can add or subtract any power of 2 (i.e. any number of the form 2^i with i ≥ 0). The goal is to determine the minimum number of operations required to make n equal to 0.

Key Insights

• The problem is equivalent to expressing n as a sum or difference of powers of 2, which connects to the idea of representing n in a signed binary form (non-adjacent form) that minimizes nonzero digits.
• When n is even, no operation is needed (other than a division by 2, which corresponds to a right shift in binary) because the least significant bit is 0.
• When n is odd, the optimal move is to either add or subtract 1 (i.e. 2^0); choosing correctly is crucial, and a known heuristic is:
  • If n equals 3 or n mod 4 is 1, subtract 1.
  • Otherwise, add 1.
• This greedy approach effectively minimizes the number of nonzero bits in the eventual signed binary representation, leading to the minimal number of operations.

Space and Time Complexity

Time Complexity: O(log n) — The number n roughly halves on even steps. Space Complexity: O(1) — Uses a fixed amount of extra space.

Solution

We employ a greedy iterative strategy:

1. Initialize a counter to track the number of operations.
2. While n is greater than 0, check if n is even. If so, divide it by 2 (simulating a right bit-shift).
3. If n is odd, decide whether to subtract or add 1 (which corresponds to subtracting or adding 2^0). Use the heuristic:
   • If n equals 3 or n modulo 4 equals 1, subtract 1.
   • Otherwise, add 1.
4. Increment the operations counter for each addition or subtraction.
5. Continue the process until n becomes 0.

This method minimizes the number of operations by effectively reducing the number of nonzero digits in the binary representation and ultimately reaching 0 with the minimum add/subtract steps needed.

Code Solutions