Maximize Happiness of Selected Children

Problem Description

You are given an array happiness of length n, and a positive integer k. There are n children standing in a queue, where the i-th child has happiness value happiness[i]. You want to select k children from these n children in k turns. In each turn, when you select a child, the happiness value of all the children that have not been selected till now decreases by 1. Note that the happiness value cannot become negative and gets decremented only if it is positive. Return the maximum sum of the happiness values of the selected children you can achieve by selecting k children.

Key Insights

• Selecting children with higher happiness values first is beneficial, as it maximizes the total happiness before decrements affect the remaining children.
• The decrements affect the happiness of unselected children, so a careful selection order is essential.
• After selecting a child, the remaining happiness values will decrease, but we can only choose k children, making the order of selection critical.

Space and Time Complexity

Time Complexity: O(n log n) - due to sorting the happiness values. Space Complexity: O(n) - for storing the sorted array.

Solution

To solve this problem, we can use a greedy approach combined with sorting:

1. Sort the happiness array in descending order: This allows us to always consider the highest available happiness value first.
2. Iterate over the top k elements: For each selected child, we calculate the current happiness value considering the decrements from previously selected children.
3. Calculate the total happiness: For the first selected child, we take its happiness directly. For the subsequent selections, we subtract the number of selections made from their happiness values (ensuring it doesn't go below zero).

This approach ensures that we maximize the sum of the happiness values of the selected children.

Code Solutions