Count of Interesting Subarrays

Problem Description

You are given a 0-indexed integer array nums, an integer modulo, and an integer k. Your task is to find the count of subarrays that are interesting. A subarray nums[l..r] is interesting if the number of indices i in the range [l, r] such that nums[i] % modulo == k satisfies cnt % modulo == k.

Key Insights

• A subarray is defined as a contiguous non-empty sequence of elements within an array.
• We need to count how many indices in the subarray meet the criteria nums[i] % modulo == k.
• The resulting count (cnt) of these indices must also satisfy the condition cnt % modulo == k.
• Efficient counting can be achieved using prefix sums and hash tables to track occurrences.

Space and Time Complexity

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

Solution

To solve the problem, we can utilize a prefix sum approach combined with a hash table to keep track of counts. The main steps are as follows:

1. Traverse the nums array while maintaining a count of how many elements satisfy the condition nums[i] % modulo == k.
2. Use a hash table to store the frequency of each count modulo modulo.
3. For each prefix count, check how many previous counts exist that would satisfy the interesting condition.
4. Increment the result based on the counts stored in the hash table.

By using this approach, we ensure that we efficiently count valid subarrays without needing to explicitly check all possible subarrays.

Code Solutions