K Divisible Elements Subarrays

Problem Description

Given an integer array nums and two integers k and p, return the number of distinct subarrays, which have at most k elements that are divisible by p.

Key Insights

• A subarray is a contiguous sequence of elements in the array.
• Two arrays are considered distinct if they differ in length or at any index.
• To count the valid subarrays, we need to keep track of how many elements in each subarray are divisible by p.
• A brute force solution could involve generating all subarrays and checking their properties, but an optimized approach is necessary for efficiency.

Space and Time Complexity

Time Complexity: O(n^2)
Space Complexity: O(n)

Solution

To solve this problem, we can use a sliding window technique combined with a set to store unique subarrays. The algorithm proceeds as follows:

1. Iterate through the array to fix the starting index of the subarray.
2. For each starting index, extend the subarray until it exceeds k elements divisible by p.
3. Use a set to keep track of unique subarrays represented as tuples (to ensure hashability).
4. Each time a new element is added to the subarray, check if it is divisible by p and update the count of divisible elements.
5. If the count exceeds k, break out of the loop.
6. Return the size of the set at the end, which represents the number of unique valid subarrays.

Code Solutions