2565. Subsequence With the Minimum Score
Description
You are given two strings s
and t
.
You are allowed to remove any number of characters from the string t
.
The score of the string is 0
if no characters are removed from the string t
, otherwise:
- Let
left
be the minimum index among all removed characters. - Let
right
be the maximum index among all removed characters.
Then the score of the string is right - left + 1
.
Return the minimum possible score to make t
a subsequence of s
.
A subsequence of a string is a new string that is formed from the original string by deleting some (can be none) of the characters without disturbing the relative positions of the remaining characters. (i.e., "ace"
is a subsequence of "abcde"
while "aec"
is not).
Example 1:
Input: s = "abacaba", t = "bzaa" Output: 1 Explanation: In this example, we remove the character "z" at index 1 (0-indexed). The string t becomes "baa" which is a subsequence of the string "abacaba" and the score is 1 - 1 + 1 = 1. It can be proven that 1 is the minimum score that we can achieve.
Example 2:
Input: s = "cde", t = "xyz" Output: 3 Explanation: In this example, we remove characters "x", "y" and "z" at indices 0, 1, and 2 (0-indexed). The string t becomes "" which is a subsequence of the string "cde" and the score is 2 - 0 + 1 = 3. It can be proven that 3 is the minimum score that we can achieve.
Constraints:
1 <= s.length, t.length <= 105
s
andt
consist of only lowercase English letters.
Solutions
Solution 1: Prefix and Suffix Preprocessing + Binary Search
According to the problem, we know that the range of the index to delete characters is [left, right]
. The optimal approach is to delete all characters within the range [left, right]
. In other words, we need to delete a substring from string $t$, so that the remaining prefix of string $t$ can match the prefix of string $s$, and the remaining suffix of string $t$ can match the suffix of string $s$, and the prefix and suffix of string $s$ do not overlap. Note that the match here refers to subsequence matching.
Therefore, we can preprocess to get arrays $f$ and $g$, where $f[i]$ represents the minimum number of characters in the prefix $t[0,..i]$ of string $t$ that match the first $[0,..f[i]]$ characters of string $s$; similarly, $g[i]$ represents the maximum number of characters in the suffix $t[i,..n-1]$ of string $t$ that match the last $[g[i],..n-1]$ characters of string $s$.
The length of the deleted characters has monotonicity. If the condition is satisfied after deleting a string of length $x$, then the condition is definitely satisfied after deleting a string of length $x+1$. Therefore, we can use the method of binary search to find the smallest length that satisfies the condition.
The time complexity is $O(n \times \log n)$, and the space complexity is $O(n)$. Where $n$ is the length of string $t$.
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