Append K Integers With Minimal Sum

Append K Integers With Minimal Sum

What is this problem about?

The "Append K Integers With Minimal Sum interview question" is a greedy math problem. You are given an array of unique positive integers and an integer k. You need to find $k$ positive integers that are not in the array and append them such that their sum is as small as possible. The result is the total sum of these $k$ appended integers.

Why is this asked in interviews?

Amazon often asks the "Append K Integers With Minimal Sum coding problem" to test a candidate's ability to optimize a mathematical calculation. While the problem sounds like you should iterate one by one, the constraints require a more efficient approach using arithmetic series and sorting. It evaluates "Math interview pattern" skills and the ability to handle large numeric results.

Algorithmic pattern used

This problem follows the Sorting and Greedy Math (Arithmetic Series) patterns.

1. Initial Goal: The $k$ smallest positive integers are simply $1$ to $k$. The sum is $(k imes (k+1)) / 2$.
2. Adjustment: Iterate through the sorted array. If an element $x$ is less than or equal to our current "threshold" (the largest number we've included in our sum), it means one of the numbers we wanted to use is already taken.
3. Replacement: Since we can't use $x$, we must skip it and "reach" for the next available smallest integer (threshold + 1).
4. Update: Add the next available number to the sum and subtract the duplicate $x$.

Example explanation

Array: [1, 4], $k = 2$.

1. Smallest 2 integers: 1, 2. Sum = 3.
2. Check array:
   • 1 is in the array. We can't use it. Replace 1 with the next smallest available, which is 3. Sum becomes 3 - 1 + 3 = 5.
   • 4 is greater than our current set {2, 3}, so it doesn't interfere. Final appended integers: [2, 3]. Sum = 5.

Common mistakes candidates make

• Brute Force: Using a while loop and a set to find the next smallest number one by one. This will time out if $k$ is $10^9$.
• Integer Overflow: Forgetting that the sum of $10^9$ integers will exceed a 32-bit integer capacity.
• Not Sorting: Trying to process the array without sorting it first, which makes it impossible to identify gaps efficiently.

Interview preparation tip

Always remember the formula for the sum of the first $n$ natural numbers. In competitive programming and interviews, "minimal sum" often points to using the smallest available numbers, which is a classic application of greedy logic.