Minimum Operations to Make Numbers Non-positive

Minimum Operations to Make Numbers Non-positive

1. What is this problem about?

This Hard-level problem involves an array of numbers and two types of attacks. A "major" attack reduces a chosen element by x and all other elements by y (where x > y). The goal is to find the minimum number of operations (attacks) required to make all elements in the array non-positive (<= 0).

2. Why is this asked in interviews?

Companies like Snowflake and Citadel ask this to test a candidate's ability to use Binary Search on Answer. This is a classic pattern where finding the direct answer is difficult, but checking if a specific number of operations T is sufficient is relatively easy. It tests your ability to identify monotonicity in the problem space.

3. Algorithmic pattern used

The primary pattern is Binary Search on the result. The possible range for the number of operations is [0, max(nums)/y + 1]. For a fixed number of operations T, every element already decreases by T * y. If an element nums[i] is still positive, it needs additional "major" damage. Each major attack provides an extra x - y damage to the chosen element. We calculate the total extra attacks needed; if they are <= T, then T operations are sufficient.

4. Example explanation

nums = [10, 12], x = 5, y = 2. Try T = 3 operations.

• Passive damage = 3 * 2 = 6.
• Remaining: 10 - 6 = 4, 12 - 6 = 6.
• Extra damage needed for 10: ceil(4 / (5-2)) = ceil(4/3) = 2 extra attacks.
• Extra damage needed for 12: ceil(6 / (5-2)) = ceil(6/3) = 2 extra attacks.
• Total extra = 4. Since 4 > 3, T=3 is not enough. Try T = 4.
• Passive damage = 8. Remaining: 2, 4.
• Extra needed: ceil(2/3) = 1, ceil(4/3) = 2.
• Total extra = 3. Since 3 <= 4, T=4 is enough.

5. Common mistakes candidates make

The most common mistake is trying to use a Greedy approach by always attacking the largest element. While this sounds intuitive, it can be difficult to implement efficiently with very large numbers. Another error is in the binary search "check" function, specifically in calculating the additional damage: you must divide by x - y, not x. Handling potential integer overflow during the calculation of T * y is also important.

6. Interview preparation tip

Binary Search on Answer is a "must-know" for Hard interview questions. Whenever a problem asks for "minimum X to satisfy a condition" and the condition becomes "easier" as X increases, binary search should be your first consideration. Practice writing the check(mid) function clearly and handling edge cases where y might be 0 or x-y might be 1.