Maximum Sum With at Most K Elements

Maximum Sum With at Most K Elements

1. What is this problem about?

The Maximum Sum With at Most K Elements problem presents a scenario where you have a matrix of numbers and a set of constraints. Each row in the matrix has a limit on how many elements you can pick from it. Additionally, there is a global limit K on the total number of elements you can select across all rows. The goal is to pick elements such that their sum is maximized while staying within both the row-level and global-level limits.

2. Why is this asked in interviews?

This "Maximum Sum With at Most K Elements interview question" is commonly asked by Microsoft and Amazon. It tests a candidate's ability to combine greedy logic with efficient data structures. It evaluates whether you can recognize that since you want the maximum sum, you should always aim to pick the largest available numbers that don't violate the row constraints, and then pick the best of those for the global limit.

3. Algorithmic pattern used

The "Array, Matrix, Sorting, Heap (Priority Queue), Greedy interview pattern" is the standard approach.

1. For each row, sort the elements in descending order and keep only the top limit_i elements allowed for that row.
2. Collect all these "best-in-row" candidates into a single large pool.
3. Sort this global pool in descending order (or use a Max-Heap).
4. Pick the top K elements from this pool and sum them up.

4. Example explanation

Matrix: Row 0: [1, 10, 5], limit=2 Row 1: [2, 3], limit=1 Global K = 2

• Best from Row 0: [10, 5].
• Best from Row 1: [3].
• Pool of candidates: [10, 5, 3].
• Sorted Pool: [10, 5, 3].
• Pick top 2: 10 + 5 = 15. Result = 15.

5. Common mistakes candidates make

In the "Maximum Sum With at Most K Elements coding problem," a common mistake is trying to use dynamic programming, which is unnecessary and far more complex than the greedy approach. Another error is picking the largest numbers globally without checking if they exceed the row-level limit first. Candidates also sometimes forget to handle cases where the total number of elements requested (K) is larger than the total number of elements available or allowed by row limits.

6. Interview preparation tip

Practice identifying "Greedy Choice" opportunities. When an optimization problem involves picking items independently (where picking an item from one row doesn't affect the availability of items in another, only the global count), greedy is usually the way to go. Mentioning the use of a Priority Queue (Heap) for selecting the top K elements is a great way to show you care about efficiency, especially if the candidate pool is very large.