Maximum Sum With Exactly K Elements

Problem Description

You are given a 0-indexed integer array nums and an integer k. Your task is to perform the following operation exactly k times in order to maximize your score:

1. Select an element m from nums.
2. Remove the selected element m from the array.
3. Add a new element with a value of m + 1 to the array.
4. Increase your score by m.

Return the maximum score you can achieve after performing the operation exactly k times.

Key Insights

• Selecting the maximum element at each step maximizes the score.
• After selecting an element, incrementing it and adding it back ensures the potential for higher scores in subsequent selections.
• A max-heap (or priority queue) can efficiently manage and retrieve the maximum element.

Space and Time Complexity

Time Complexity: O(k log n), where n is the number of elements in nums. Each of the k operations involves removing the max element and inserting the incremented value, both of which take O(log n) time in a heap.

Space Complexity: O(n), for storing the elements in the heap.

Solution

To solve the problem, we use a max-heap (priority queue) to efficiently track the maximum element in the array. The algorithm involves:

1. Initializing a max-heap with the elements from nums.
2. Repeating the operation k times:
   • Extract the maximum element from the heap.
   • Add this element to the score.
   • Increment the element by 1 and push it back into the heap.
3. Return the accumulated score.

Code Solutions