Maximum Sum With Exactly K Elements

Maximum Sum With Exactly K Elements

1. What is this problem about?

The Maximum Sum With Exactly K Elements problem is a straightforward greedy challenge. You are given an array of integers and an integer k. You need to perform a specific operation exactly k times: pick the maximum element x in the array, add it to your score, and then replace x with x + 1 in the array. Your goal is to find the maximum possible total score after k operations.

2. Why is this asked in interviews?

This "Maximum Sum With Exactly K Elements interview question" is an "EASY" problem often used by Meta and Amazon for early-round screenings or as a "warm-up" during a coding session. It tests a candidate's ability to follow a greedy strategy and recognize a simple arithmetic progression. Even though it's simple, it requires careful implementation to avoid unnecessary complexity (like re-sorting the array in every step).

3. Algorithmic pattern used

The "Array, Greedy interview pattern" is clear: always pick the largest number. Since adding 1 to the largest number makes it even larger, that same element will always be the maximum for all k operations. Therefore, you just need to find the initial maximum value M. The total sum will be the sum of an arithmetic progression: M + (M+1) + (M+2) + ... + (M + k-1). This can be calculated in O(1) time after finding the initial maximum in O(N).

4. Example explanation

Array: [1, 2, 5], k = 3

• Initial Max = 5.
• Op 1: Pick 5. Score = 5. Array becomes [1, 2, 6].
• Op 2: Pick 6. Score = 5 + 6 = 11. Array becomes [1, 2, 7].
• Op 3: Pick 7. Score = 11 + 7 = 18. Array becomes [1, 2, 8]. Total Score = 18. Alternatively, using the formula: 5 + 6 + 7 = 18.

5. Common mistakes candidates make

In the "Maximum Sum With Exactly K Elements coding problem," the most common mistake is using a loop to find the max or a priority queue in every iteration, which makes the solution O(k log N) or O(k*N). While it passes for small k, it's inefficient. A more subtle mistake is failing to handle potential integer overflow if M and k are very large, though the formula (k * (2*M + k - 1)) / 2 is generally safe in languages with large integer support.

6. Interview preparation tip

Whenever you see a problem that involves repeating the same greedy choice multiple times, ask yourself: "Will the same element still be the best choice after the operation?" If the answer is yes (as in this case), you can use a mathematical formula to jump directly to the answer. This demonstrates a high level of analytical skill and is much more impressive to an interviewer than a simple loop.