Minimum Operations to Exceed Threshold Value II

Problem Description

You are given a 0-indexed integer array nums, and an integer k. You are allowed to perform some operations on nums, where in a single operation, you can:

1. Select the two smallest integers x and y from nums.
2. Remove x and y from nums.
3. Insert (min(x, y) * 2 + max(x, y)) at any position in the array.

Return the minimum number of operations needed so that all elements of the array are greater than or equal to k.

Key Insights

• The operation involves combining the two smallest elements, which can potentially create a larger element.
• A priority queue (min-heap) is suitable for efficiently retrieving the smallest elements.
• The goal is to keep performing operations until all elements meet or exceed the threshold k.
• Each operation reduces the size of the array, so we need to carefully track how many operations are performed.

Space and Time Complexity

Time Complexity: O(n log n) - Each operation involves popping from the heap and inserting a new element, which takes logarithmic time relative to the number of elements in the heap.

Space Complexity: O(n) - We use a heap to store the elements of the array.

Solution

To solve the problem, we will use a min-heap to efficiently retrieve and combine the two smallest elements in the array. The algorithm proceeds as follows:

1. Initialize a min-heap with the elements of nums.
2. While the smallest element in the heap is less than k, perform the following:
   • Extract the two smallest elements x and y.
   • Compute the new value to insert: min(x, y) * 2 + max(x, y).
   • Insert this new value back into the heap.
   • Increment the operation count.
3. Once all elements in the heap are greater than or equal to k, return the operation count.

Code Solutions