K Divisible Elements Subarrays

K Divisible Elements Subarrays

1. What is this problem about?

The K Divisible Elements Subarrays interview question asks you to count the number of distinct contiguous subarrays that contain at most k elements divisible by a given integer p. Two subarrays are considered distinct if they have different lengths or if they differ in at least one position's value.

2. Why is this asked in interviews?

Companies like Uber and Amazon use the K Divisible Elements coding problem to test a candidate's mastery of String Matching and Rolling Hash techniques. It’s not enough to count subarrays; you must deduplicate them. This evaluation tests whether you can efficiently represent a subarray (e.g., using a Trie or a Set of strings/hashes) to avoid $O(N^3)$ complexity. It’s a core Hash Table interview pattern.

3. Algorithmic pattern used

This problem follows the Sliding Window and Deduplication pattern.

1. Iterate All Windows: Use a nested loop to identify all subarrays nums[i...j].
2. Counting Rule: For each subarray, keep a running count of how many elements are divisible by p. If the count exceeds k, stop expanding from the current start index i.
3. Uniqueness: Use a Set<List<Integer>> or a Trie to store seen subarrays. A Trie is more memory-efficient for long overlapping sequences.
4. Hashing: Alternatively, use a rolling hash to represent each subarray as a unique long integer for faster deduplication.

4. Example explanation

nums = [2, 3, 3, 2, 2], k = 2, p = 2.

• Divisible by 2 are: 2, 2, 2.
• Subarray [2, 3, 3] has one '2'. Valid.
• Subarray [2, 3, 3, 2] has two '2's. Valid.
• Subarray [2, 3, 3, 2, 2] has three '2's. Invalid (Exceeds $k=2$).
• Subarray [3, 3] is valid. The result is the count of all unique valid subarrays found.

5. Common mistakes candidates make

• Not Deduplicating: Simply counting all valid subarrays without checking if they are distinct.
• $O(N^3)$ String Conversion: Converting every subarray to a string to put in a set, which is slow and uses massive memory.
• Trie Implementation: Errors in inserting into the Trie or not realizing that each node in the Trie represents a unique subarray.

6. Interview preparation tip

Practice using a Trie for subarray deduplication. Each path from the root to a node corresponds to a unique contiguous segment of the original array. This is a powerful Design interview pattern for sequence-based problems.