Maximum Sum With at Most K Elements

Problem Description

You are given a 2D integer matrix grid of size n x m, an integer array limits of length n, and an integer k. The task is to find the maximum sum of at most k elements from the matrix grid such that:

• The number of elements taken from the ith row of grid does not exceed limits[i].

Return the maximum sum.

Key Insights

• The problem requires selecting elements from a 2D matrix while adhering to row-specific limits.
• A greedy approach can be applied by selecting the largest elements from each row, constrained by the limits.
• A max-heap (or priority queue) can efficiently manage the largest selected elements across rows.
• It's important to consider the total number of elements selected should not exceed k.

Space and Time Complexity

Time Complexity: O(n * m log m)
Space Complexity: O(n * m)

Solution

To solve this problem, we can follow these steps:

1. For each row in the grid, sort the elements in descending order.
2. For each row, take the top elements according to the corresponding limit and store them in a max-heap.
3. Continuously extract the largest elements from the heap until we have selected k elements or run out of elements.
4. Sum the extracted elements to get the maximum sum.

This approach utilizes sorting for each row and a priority queue to maintain the largest elements efficiently.

Code Solutions