Find Minimum Cost to Remove Array Elements

Problem Description

You are given an integer array nums. Your task is to remove all elements from the array by performing one of the following operations at each step until nums is empty:

• Choose any two elements from the first three elements of nums and remove them. The cost of this operation is the maximum of the two elements removed.
• If fewer than three elements remain in nums, remove all the remaining elements in a single operation. The cost of this operation is the maximum of the remaining elements.

Return the minimum cost required to remove all the elements.

Key Insights

• The operation allows removing two elements from the first three, meaning that the choice of elements can greatly affect the total cost.
• The strategy should focus on minimizing the maximum value chosen in each step.
• If the size of the array is less than three, the remaining elements must be removed together, incurring a cost equal to the maximum of those elements.

Space and Time Complexity

Time Complexity: O(n)
Space Complexity: O(1)

Solution

To solve the problem, we can use a greedy approach combined with a priority queue (or max-heap) to always choose the maximum elements efficiently. The algorithm involves the following steps:

1. While there are three or more elements in the array, repeatedly select the two highest values from the first three elements and remove them, adding the maximum of these two values to the total cost.
2. Once fewer than three elements remain, remove all of them in one operation, adding the maximum of these remaining elements to the total cost.
3. Return the accumulated cost as the minimum cost required to remove all elements.

This approach ensures that at each step, we are selecting the optimal elements to minimize the overall cost.

Code Solutions