How to Solve Minimum Operations to Exceed Threshold Value II

This problem requires performing operations on an array to ensure all elements exceed a given threshold k. Use a min-heap to efficiently track and combine the two smallest elements repeatedly until the condition is met. The solution guarantees the minimum number of operations while maintaining optimal time complexity with the heap.

Problem Statement

Given a 0-indexed array of integers nums and an integer k, determine the minimum number of operations required to make every element in nums greater than or equal to k. Each operation involves removing the two smallest elements and inserting a combined value back into the array using a defined formula. You can only perform operations when nums has at least two elements.

Return the minimum number of operations needed to satisfy the threshold requirement. If the array already meets the condition or can reach it through repeated operations, compute the exact count. Ensure your approach efficiently handles large arrays with up to 2 * 10^5 elements.

Examples

Example 1

Input: nums = [2,11,10,1,3], k = 10

Output: 2

At this stage, all the elements of nums are greater than or equal to 10 so we can stop. It can be shown that 2 is the minimum number of operations needed so that all elements of the array are greater than or equal to 10.

Example 2

Input: nums = [1,1,2,4,9], k = 20

Output: 4

At this stage, all the elements of nums are greater than 20 so we can stop. It can be shown that 4 is the minimum number of operations needed so that all elements of the array are greater than or equal to 20.

Constraints

• 2 <= nums.length <= 2 * 105
• 1 <= nums[i] <= 109
• 1 <= k <= 109
• The input is generated such that an answer always exists. That is, after performing some number of operations, all elements of the array are greater than or equal to k.

Solution Approach

Use a Min-Heap to Track Minimum Elements

Insert all elements of nums into a min-heap. This allows quick access to the two smallest values, which is necessary for each operation. The heap ensures efficient extraction and insertion in O(log N) time.

Simulate Operations Until Threshold

Repeatedly pop the two smallest elements from the heap, combine them using the operation formula, and push the result back into the heap. Increment a counter for each operation until all elements in the heap are at least k.

Handle Edge Cases and Verify Completion

Check if the smallest element is already >= k before performing operations. If only one element remains below k, ensure the operation sequence can still achieve the threshold. Return the operation count once all elements meet the condition.

Complexity Analysis

Time complexity is O(N log N) because each heap operation (pop and push) costs O(log N) and we perform up to N operations. Space complexity is O(N) to store all array elements in the heap.

What Interviewers Usually Probe

• Does the candidate correctly identify the need for a min-heap to track smallest elements?
• Can they simulate operations efficiently without iterating over the entire array each time?
• Do they consider edge cases where repeated operations barely meet the threshold?

Common Pitfalls or Variants

Common pitfalls

• Trying to sort the array repeatedly instead of using a heap, which increases time complexity.
• Failing to check the array length before performing an operation, leading to invalid operations.
• Incorrectly combining elements or forgetting to increment the operation counter properly.

Follow-up variants

• Change the operation formula to include multiplication instead of addition and test heap behavior.
• Require that only a subset of the array must exceed k instead of all elements.
• Apply the same heap simulation approach on a dynamic stream of elements rather than a static array.

How GhostInterview Helps

• Automatically structures the min-heap approach for the Minimum Operations to Exceed Threshold Value II problem.
• Provides step-by-step operation simulation while tracking the exact number of moves needed.
• Highlights potential edge cases and failure modes that arise from small or large values in the array.

Topic Pages

• Array
• Heap (Priority Queue)
• Simulation

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FAQ

What is the best data structure for Minimum Operations to Exceed Threshold Value II?

A min-heap efficiently tracks the two smallest elements needed for each operation to meet the threshold condition.

Can I solve this problem without a heap?

Technically yes by sorting repeatedly, but it is much less efficient and can exceed time limits for large arrays.

What if the array has exactly two elements?

You can perform the operation once if needed, then check if both elements meet or exceed k. The heap still works correctly.

Does the operation formula affect the solution pattern?

Yes, the array plus heap pattern depends on combining the smallest elements using the specified formula. Different formulas require careful adjustment.

What is the failure mode to watch for?

The main failure is attempting operations when fewer than two elements remain or miscomputing the combined value, leading to incorrect counts.