Count K-Subsequences of a String With Maximum Beauty

Count K-Subsequences of a String With Maximum Beauty

What is this problem about?

The "Count K-Subsequences of a String With Maximum Beauty" problem is a complex combinatorial challenge. You are given a string and an integer $k$. A "k-subsequence" is a subsequence of length $k$ where all characters are unique. The "beauty" of such a subsequence is the sum of the frequencies of its characters in the original string. You need to find the number of unique k-subsequences that achieve the maximum possible beauty.

Why is this asked in interviews?

Google asks this problem to evaluate a candidate's ability to combine multiple concepts: frequency counting, greedy selection, and combinatorics. It’s not just about finding the highest values; it's about counting how many ways those values can be formed. This tests analytical depth and the ability to handle large numerical results using modulo arithmetic.

Algorithmic pattern used

This problem uses a Greedy approach combined with Combinatorics (specifically, the multiplication principle and combinations).

1. Count the frequency of each character in the original string.
2. If the number of unique characters is less than $k$, it's impossible to form a k-subsequence (return 0).
3. Sort the frequencies in descending order. The maximum beauty is the sum of the top $k$ frequencies.
4. To count the ways:
   • Identify the value of the $k$-th largest frequency (let's call it $f$).
   • Count how many characters have a frequency strictly greater than $f$ (let's call this $count_{gt}$).
   • Count how many characters have frequency exactly $f$ (let's call this $count_{eq}$).
   • You must pick all $count_{gt}$ characters and $(k - count_{gt})$ characters from the $count_{eq}$ pool.
   • Use $\binom{count_{eq}}{k - count_{gt}} imes f^{(k - count_{gt})} imes \prod ( ext{frequencies of characters } > f)$ to get the result.

Example explanation

String: abbcccddd, $k = 2$.

• Frequencies: a:1, b:2, c:3, d:3. Sorted: [3, 3, 2, 1].
• Top $k=2$ frequencies are 3 and 3. Max beauty = 6.
• Characters with frequency 3: 'c' and 'd'.
• We need to pick 2 characters from the 2 available with frequency 3.
• Ways to pick 'c': 3 options (any of the three 'c's).
• Ways to pick 'd': 3 options.
• Total ways: $\binom{2}{2} imes 3 imes 3 = 1 imes 9 = 9$.

Common mistakes candidates make

A common mistake is forgetting that choosing a character like 'b' (frequency 2) contributes 2 to the beauty but provides 2 different subsequences because you can pick either instance of 'b'. Another error is failing to use modulo arithmetic at each multiplication step, leading to integer overflow. Many also struggle with the combinatorial part when multiple characters share the same "threshold" frequency.

Interview preparation tip

When counting "maximum" something, always separate the task into two steps: first, find what the maximum value is (usually greedy), and second, count the ways to achieve it. For counting, always group identical values together and use the combination formula ($nCr$).