Rearrange Array to Maximize Prefix Score

Problem Description

You are given a 0-indexed integer array nums. You can rearrange the elements of nums to any order (including the given order). Let prefix be the array containing the prefix sums of nums after rearranging it. In other words, prefix[i] is the sum of the elements from 0 to i in nums after rearranging it. The score of nums is the number of positive integers in the array prefix. Return the maximum score you can achieve.

Key Insights

• The order of elements in the array greatly influences the prefix sums.
• To maximize the number of positive prefix sums, larger positive numbers should be placed at the start of the array.
• Negative numbers at the end of the array will have less impact on the prefix sums.
• Sorting the array in descending order allows for maximizing the prefix sum at each index.

Space and Time Complexity

Time Complexity: O(n log n) - due to sorting the array.
Space Complexity: O(1) - we use a constant amount of additional space.

Solution

To solve the problem, we can follow these steps:

1. Sort the array in descending order.
2. Initialize a variable to keep track of the current prefix sum and the score.
3. Iterate through the sorted array, updating the prefix sum and incrementing the score whenever the prefix sum is positive.
4. Return the score, which represents the maximum number of positive prefix sums achievable.

The main data structure used is an array, and the algorithmic approach is a greedy strategy combined with sorting.

Code Solutions