Minimum Operations to Exceed Threshold Value II

Minimum Operations to Exceed Threshold Value II

1. What is this problem about?

This is a more complex version of the threshold problem. You are given an array and a threshold k. Instead of just removing elements, you perform a specific operation: take the two smallest elements x and y, and replace them with a single new element calculated as min(x, y) * 2 + max(x, y). You repeat this until every element in the array is at least k.

2. Why is this asked in interviews?

This "Minimum Operations to Exceed Threshold Value II interview question" is popular at Meta, Google, and Amazon because it requires an efficient way to repeatedly find the smallest elements. It's a classic application of a Priority Queue (Min-Heap). It tests if a candidate can handle simulation problems where the data changes dynamically.

3. Algorithmic pattern used

The primary pattern is Simulation using a Min-Heap. You insert all elements into a min-heap. While the smallest element in the heap is less than k, you pop the two smallest elements, calculate the new value, and push it back. You keep a count of how many such operations you perform.

4. Example explanation

• Array: [1, 1, 2, 4, 9], k = 20
• Smallest are 1 and 1. New value: 1 * 2 + 1 = 3. Heap: [2, 3, 4, 9].
• Smallest are 2 and 3. New value: 2 * 2 + 3 = 7. Heap: [4, 7, 9].
• Smallest are 4 and 7. New value: 4 * 2 + 7 = 15. Heap: [9, 15].
• Smallest are 9 and 15. New value: 9 * 2 + 15 = 33. Heap: [33]. Now 33 >= 20. Total operations: 4.

5. Common mistakes candidates make

The most common mistake is attempting to sort the array in every step, which leads to O(n² log n) complexity. A Min-Heap is required to keep it efficient at O(n log n). Another error is forgetting to handle the case where the calculation min(x, y) * 2 + max(x, y) might overflow standard integer types, although many languages handle this gracefully.

6. Interview preparation tip

Whenever a problem involves "repeatedly picking the smallest/largest elements," a Priority Queue (Heap) is almost certainly the correct data structure. Practice implementing min-heaps and max-heaps to solve these types of simulation problems quickly.