Subsequence With the Minimum Score

Subsequence With the Minimum Score

What is this problem about?

"Subsequence With the Minimum Score" is a hard string optimization problem. You are given two strings, s and t. You are allowed to remove a contiguous substring from t to make the remaining parts of t a subsequence of s. The "score" of this operation is the length of the removed substring. Your goal is to find the minimum possible score. This problem requires finding the best "cut" in string t such that the prefix and suffix of t can be found as a subsequence within s.

Why is this asked in interviews?

This question is a favorite at DoorDash and other high-growth tech companies because it tests a candidate's ability to combine multiple string techniques. It is not a standard subsequence problem; it requires pre-calculating the positions of prefixes and suffixes. It evaluates the ability to use "Prefix and Suffix arrays" to solve an optimization problem in linear time, which is a sophisticated approach compared to basic string manipulation.

Algorithmic pattern used

The problem is solved using Two Pointers and Prefix/Suffix Arrays.

1. Create a prefix array where prefix[i] is the index of the last character in s that matches the prefix of t of length i+1.
2. Create a suffix array where suffix[i] is the index of the first character in s that matches the suffix of t of length len(t)-i.
3. Once you have these arrays, you can use a sliding window or a binary search over the possible scores to find the minimum length of the removed substring in t that allows the remaining parts to fit as a subsequence in s.

Example explanation (use your own example)

s = "abacaba", t = "asdfaba"

1. The prefix of t is "asdf...". Only "a" is a subsequence in s.
2. The suffix of t is "...aba". This is a subsequence in s (at indices 4, 5, 6).
3. If we remove "sdf", the remaining part of t is "a" + "aba" = "aaba".
4. Check if "aaba" is a subsequence of "abacaba". Yes.
5. The score (length of "sdf") is 3. We try to find if a smaller score is possible.

Common mistakes candidates make

A common error is trying to solve this using dynamic programming, which would result in O(N*M) complexity and likely time out. Another mistake is not correctly handling the case where the prefix and suffix of t might overlap in their mapping to s. Candidates also often struggle with the indices when building the prefix and suffix arrays, especially when iterating backwards for the suffix.

Interview preparation tip

When preparing for the Subsequence With the Minimum Score interview question, practice the "Prefix/Suffix Pre-computation" pattern. It's used in many hard string problems. Being able to explain why you are pre-calculating the "earliest possible" and "latest possible" positions for characters will show the interviewer that you have a structured approach to subsequence problems.