Count Special Subsequences

Count Special Subsequences

What is this problem about?

The "Count Special Subsequences interview question" is an optimization problem focused on ratios and products. You are given an integer array and need to find the number of distinct quadruplets of indices $(p, q, r, s)$ such that $p < q < r < s$ and the product of elements at those indices satisfies a specific ratio: nums[p] / nums[q] == nums[s] / nums[r]. This equation can be rewritten as nums[p] * nums[r] == nums[s] * nums[q].

Why is this asked in interviews?

Companies like Google and Bloomberg ask the "Count Special Subsequences coding problem" to evaluate a candidate's ability to use "Hash Table interview pattern" techniques to simplify complex constraints. A brute-force approach would be $O(N^4)$, which is too slow. The challenge is to find an $O(N^2)$ solution by splitting the problem into two halves and storing intermediate results in a map.

Algorithmic pattern used

This problem follows the Meet-in-the-Middle and Hash Map patterns.

1. Algebraic Transformation: Rewrite the ratio nums[p] / nums[q] == nums[s] / nums[r] as (nums[p] / nums[q]) == (nums[s] / nums[r]).
2. Iterative Scan: Fix the gap between $q$ and $r$. As you move the $q$ and $r$ pointers, you maintain a count of all valid nums[p] / nums[q] seen so far.
3. Map Storage: Use a hash map to store the frequencies of the ratios. To avoid floating-point issues, store the ratios as reduced fractions (using GCD) or simply use the cross-product logic if memory allows.
4. Counting: For each new pair $(r, s)$, calculate the ratio nums[s] / nums[r] and check how many times this ratio occurred in the (p, q) pairs processed earlier.

Example explanation

Array: [1, 2, 3, 4, 3, 2, 1]

• Fix a split point between indices 2 and 3 ($q=2, r=3$).
• Pairs $(p, q)$ to the left: $(0, 1), (0, 2), (1, 2)$. Ratios: $1/2, 1/3, 2/3$.
• Pairs $(r, s)$ to the right: $(3, 4), (3, 5), (3, 6)...$.
• If nums[s] / nums[r] is $1/2$ (e.g., $nums[6]=1, nums[5]=2$), we have a match. Result: Increment the total count for each matching pair.

Common mistakes candidates make

• Floating Point Errors: Using double for ratios, which can lead to precision issues. It's better to store fractions as a pair of integers (numerator, denominator) simplified by their GCD.
• Index Management: Failing to correctly maintain the split between the "left" pairs and "right" pairs as the loop progresses.
• Brute Force: Checking all $O(N^4)$ quadruplets directly.

Interview preparation tip

Whenever you see a four-variable equation ($A/B = C/D$ or $A+B = C+D$), think about grouping them into two pairs ($A/B$ and $C/D$). This transformation is the key to reducing the dimensionality of many "Enumeration interview pattern" problems.