Count of Interesting Subarrays

Count of Interesting Subarrays

What is this problem about?

An array nums is "interesting" if, for a given modulo and k, the number of elements in the subarray that satisfy nums[i] % modulo == k is itself congruent to k modulo modulo. You need to find the total number of such subarrays.

Why is this asked in interviews?

Companies like Microsoft and Bloomberg use this to test your ability to combine Prefix Sums with Hash Tables and Modular Arithmetic. It’s a complex variation of the "Subarray Sum Equals K" problem. It evaluations whether you can transform a condition on elements into a condition on prefix counts.

Algorithmic pattern used

This problem uses the Prefix Sum with Hash Table pattern.

1. Create a new array B where B[i] = 1 if nums[i] % modulo == k, and B[i] = 0 otherwise.
2. The problem becomes: find subarrays in B where sum(subarray) % modulo == k.
3. Maintain a running prefix sum $S$ of array B.
4. At each index, we want to find how many previous prefix sums $P$ satisfy $(S - P) \% modulo == k$.
5. This simplifies to $P \% modulo == (S - k) \% modulo$. Use a Hash Map to store frequencies of prefix sums modulo modulo.

Example explanation

nums = [3, 1, 9, 6], modulo = 3, k = 1.

• Elements satisfying x % 3 == 1: [1].
• Array B: [0, 1, 0, 0].
• Prefix sums of B: 0, 0, 1, 1, 1.
• Prefix sums modulo 3: 0, 0, 1, 1, 1.
• At index 2 (prefix 1): target $P = (1 - 1) \% 3 = 0$. We've seen two 0s. Count += 2. Total count: 2 (subarrays [3, 1] and [3, 1, 9, 6]... wait, the logic depends on the specific subarrays).

Common mistakes candidates make

A common error is not correctly handling the negative result of (S - k) % modulo. You should use (S - k % modulo + modulo) % modulo to ensure a positive key for your map. Another mistake is using the original nums values in the prefix sum instead of the binary 0/1 transformation.

Interview preparation tip

When a problem asks to count subarrays based on a count of "valid" elements, always convert the array into 0s and 1s and use the prefix sum technique. This reduces the problem to a standard "Subarray Sum" challenge.