Minimum Operations to Make the Integer Zero

Minimum Operations to Make the Integer Zero

1. What is this problem about?

This "Medium" brainteaser involves two integers, num1 and num2. In one operation, you can subtract (2^i + num2) from num1, where i is any non-negative integer. You want to find the minimum number of operations required to make num1 exactly zero. If it is impossible, return -1.

2. Why is this asked in interviews?

Meta and Amazon ask this to test bit manipulation skills and the ability to find a mathematical boundary. It's not a standard BFS/DFS problem because the choice of i is huge. Instead, it tests if you can rephrase the equation. If we use k operations, we have: num1 - k * num2 = sum(2^i_j) for j from 1 to k. This means the value num1 - k * num2 must be representable as a sum of exactly k powers of 2.

3. Algorithmic pattern used

The pattern is Enumeration and Bit Manipulation. You iterate through the possible number of operations k starting from 1. For a fixed k, let target = num1 - k * num2. For target to be representable as a sum of k powers of two, two conditions must be met:

1. target must be >= k (because the smallest sum of k powers of 2 is k * 2^0 = k).
2. The number of set bits (1-bits) in target must be <= k (because each set bit can be split into smaller powers of 2 until we have exactly k items).

4. Example explanation

num1 = 85, num2 = -5.

• k = 1: target = 85 - 1*(-5) = 90. Set bits in 90 (1011010) = 4. Since 4 > 1, k=1 fails.
• k = 2: target = 85 - 2*(-5) = 95. Set bits in 95 (1011111) = 6. Since 6 > 2, k=2 fails.
• ...
• k = 5: target = 85 - 5*(-5) = 110. Set bits in 110 (1101110) = 5.
  • Condition 1: 110 >= 5 (True).
  • Condition 2: Set bits (5) <= 5 (True). Result = 5.

5. Common mistakes candidates make

Candidates often try to use dynamic programming or recursion, which is difficult due to the num2 factor. They also frequently forget the first condition (target >= k). Some might not realize that the maximum value of k is relatively small (around 40-60), so a simple loop is sufficient. Miscalculating the number of set bits (using a non-efficient method) can also be a minor issue.

6. Interview preparation tip

When a problem involves powers of 2, always check if it can be solved by looking at bit counts (population count). Learn the properties of binary representation: a number X can be represented as a sum of N powers of 2 if and only if popcount(X) <= N <= X. This is a very useful property for "minimum operations" problems involving bits.