<p>You are given two strings <code>s</code> and <code>t</code>.</p>

<p>You are allowed to remove any number of characters from the string <code>t</code>.</p>

<p>The score of the string is <code>0</code> if no characters are removed from the string <code>t</code>, otherwise:</p>

<ul>
	<li>Let <code>left</code> be the minimum index among all removed characters.</li>
	<li>Let <code>right</code> be the maximum index among all removed characters.</li>
</ul>

<p>Then the score of the string is <code>right - left + 1</code>.</p>

<p>Return <em>the minimum possible score to make </em><code>t</code><em>&nbsp;a subsequence of </em><code>s</code><em>.</em></p>

<p>A <strong>subsequence</strong> 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., <code>&quot;ace&quot;</code> is a subsequence of <code>&quot;<u>a</u>b<u>c</u>d<u>e</u>&quot;</code> while <code>&quot;aec&quot;</code> is not).</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> s = &quot;abacaba&quot;, t = &quot;bzaa&quot;
<strong>Output:</strong> 1
<strong>Explanation:</strong> In this example, we remove the character &quot;z&quot; at index 1 (0-indexed).
The string t becomes &quot;baa&quot; which is a subsequence of the string &quot;abacaba&quot; and the score is 1 - 1 + 1 = 1.
It can be proven that 1 is the minimum score that we can achieve.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> s = &quot;cde&quot;, t = &quot;xyz&quot;
<strong>Output:</strong> 3
<strong>Explanation:</strong> In this example, we remove characters &quot;x&quot;, &quot;y&quot; and &quot;z&quot; at indices 0, 1, and 2 (0-indexed).
The string t becomes &quot;&quot; which is a subsequence of the string &quot;cde&quot; and the score is 2 - 0 + 1 = 3.
It can be proven that 3 is the minimum score that we can achieve.
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= s.length, t.length &lt;= 10<sup>5</sup></code></li>
	<li><code>s</code> and <code>t</code> consist of only lowercase English letters.</li>
</ul>

class Solution:
    def minimumScore(self, s: str, t: str) -> int:
        m, n = len(s), len(t)
        
        # Initialize dp arrays
        dp_left = [0] * (m + 1)
        dp_right = [n] * (m + 1)
        
        # dp_left : Using i chars in s, how many of t can we cover
        j = 0
        for i in range(m):
            if j < n and s[i] == t[j]:
                j += 1
            dp_left[i + 1] = j
        
        # dp_right : Using i suffix chars in s, how many suffix chars of t can we cover
        j = n - 1
        for i in range(m - 1, -1, -1):
            if j >= 0 and s[i] == t[j]:
                j -= 1
            dp_right[i] = j + 1

        # Find minimum score
        # min_score = min(dp_left)
        min_score = n
        for i in range(m + 1):
            if dp_left[i] == n:
                return 0  # t is already a subsequence of s
            if dp_right[i] > dp_left[i]:
                # left covers dp_left[i] chars prefix
                # right covers (n - 1 - dp_right[i] + 1) chars suffix
                # rem = n - (dp_left[i] + n - dp_right[i]) = dp_right[i] - dp_left[i]
                min_score = min(min_score, dp_right[i] - dp_left[i])
        
        return min_score

24900