Smallest Missing Non-negative Integer After Operations

Smallest Missing Non-negative Integer After Operations

What is this problem about?

The "Smallest Missing Non-negative Integer After Operations" interview question involves modular arithmetic and the MEX (Minimum Excluded value) concept. You are given an array and a value value. You can add or subtract value from any element any number of times. Your goal is to find the smallest non-negative integer (MEX) that cannot be formed after these operations. This "Smallest Missing Non-negative Integer After Operations coding problem" asks you to optimize the array to fill the smallest possible gaps.

Why is this asked in interviews?

Companies like Microsoft and Atlassian ask this to test your understanding of congruence classes. Since adding/subtracting value doesn't change an element's remainder when divided by value, the problem is really about how many elements you have in each "remainder bucket". It evaluates your ability to use hash maps or frequency arrays to solve optimization problems.

Algorithmic pattern used

This problem follows the "Math and Hash Table (Frequency Map) interview pattern".

1. Every number x can be converted to any other number y if x % value == y % value (handling negative results carefully).
2. Count the frequency of each remainder: count[x % value].
3. To form the number i, you need to have at least one element with the remainder i % value.
4. The number of times you can form a number with remainder r depends on count[r]. Specifically, you can form r, r + value, r + 2*value, ... up to r + (count[r]-1)*value.
5. The answer is the first integer i such that we have already used up all available elements with remainder i % value.

Example explanation

Array: [1, 1, 2], value = 3.

1. Remainders: 1 % 3 = 1 (appears twice), 2 % 3 = 2 (appears once).
2. Can we form 0? 0 % 3 = 0. Count is 0. No. MEX is 0. Array: [0, 1, 2, 1], value = 3.
3. Remainders: 0:1, 1:2, 2:1.
4. Form 0: 0 % 3 = 0. Have one '0'. OK.
5. Form 1: 1 % 3 = 1. Have two '1's. OK.
6. Form 2: 2 % 3 = 2. Have one '2'. OK.
7. Form 3: 3 % 3 = 0. Used the only '0' remainder already. NO. MEX is 3.

Common mistakes candidates make

A frequent error is not handling negative numbers correctly when calculating the modulo (the result should always be in the range [0, value-1]). Another mistake is trying to actually perform the additions/subtractions instead of just counting the remainders. Some candidates also struggle to realize that the MEX can be larger than the array size.

Interview preparation tip

The MEX of an array is always between 0 and the size of the array. In this "Smallest Missing Non-negative Integer After Operations interview question", the key is realizing that you should "spend" your remainder counts on the smallest possible numbers first.