Minimum Increment to Make Array Unique

Minimum Increment to Make Array Unique

1. What is this problem about?

The "Minimum Increment to Make Array Unique" coding problem asks you to take an array of integers that may contain duplicate values and transform it into an array where every element is unique. The only operation allowed is to pick any element and increment its value by 1. The goal is to find the minimum number of such operations required to achieve uniqueness.

For instance, if you have two 2s in your array, you cannot just leave them. You must increase one of them. If you increase it to 3, but a 3 already exists in the array, you must increase it further. The challenge lies in doing this efficiently so that the total number of "steps" (increments) is as small as possible.

2. Why is this asked in interviews?

This question is a favorite for companies like Meta, Amazon, and Microsoft because it tests a candidate's grasp of Sorting and Greedy algorithms. While the problem seems simple at first glance, a brute-force approach (checking and incrementing repeatedly) will be too slow for large inputs.

Interviewer's look for:

• Efficiency Awareness: Can the candidate avoid $O(n^2)$ solutions?
• Sorting Intuition: Does the candidate realize that ordering the data simplifies the duplicate-finding process?
• Mathematical Logic: Can the candidate track the "expected next value" to calculate increments in a single pass?

3. Algorithmic pattern used

There are two primary patterns used to solve the Minimum Increment to Make Array Unique interview question:

1. Sorting (Greedy): By sorting the array first, duplicates are placed next to each other. We can then iterate through the array and ensure each element is at least one greater than the previous one. If it's not, we increment it to previous + 1 and add the difference to our total count.
2. Counting (Frequency Array): If the range of numbers is limited, we can count the frequency of each number. We then iterate through the frequency array; if a number appears more than once, we "carry over" the extras to the next possible value.

4. Example explanation

Consider the array [3, 2, 1, 2, 1, 7]. First, let's sort it: [1, 1, 2, 2, 3, 7].

1. The first 1 is fine.
2. The second 1 is a duplicate. It needs to be at least 1 + 1 = 2. So we increment it to 2. Operations: $2 - 1 = 1$. Array: [1, 2, 2, 2, 3, 7].
3. The next element is 2, but the previous was 2. It must be at least 2 + 1 = 3. Operations: $3 - 2 = 1$. Array: [1, 2, 3, 2, 3, 7]. (Wait, let's track the "needed" value properly). Actually, a better way to track: Sorted: [1, 1, 2, 2, 3, 7]

• Index 1: 1. Next must be $\ge 2$.
• Index 2: 1. Needs to be 2. Operations: 1. Current value is now 2. Next must be $\ge 3$.
• Index 3: 2. Needs to be 3. Operations: 1. Current value is now 3. Next must be $\ge 4$.
• Index 4: 2. Needs to be 4. Operations: 2. Current value is now 4. Next must be $\ge 5$.
• Index 5: 3. Needs to be 5. Operations: 2. Current value is now 5. Next must be $\ge 6$.
• Index 6: 7. It's already $\ge 6$, so no operations needed. Total operations: $1 + 1 + 2 + 2 = 6$.

5. Common mistakes candidates make

• Repeatedly Scanning: Some candidates try to increment a value and then re-scan the array to see if the new value is unique. This leads to $O(n^2)$ or worse performance.
• Ignoring Constraints: Not noticing that the numbers can grow quite large after many increments, potentially affecting space if using a frequency array.
• Sorting Overhead: In some very specific high-performance scenarios, sorting might be slower than a linear counting approach, but for most interviews, sorting is the expected solution.

6. Interview preparation tip

Master the "Sort and Sweep" technique. It’s a versatile pattern where sorting data allows you to solve a complex uniqueness or interval problem in a single linear pass afterward. Always ask about the range of input values; if the range is small, a counting sort or frequency array might be a more "impressive" answer than a standard sort.