<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:
        """
        t[0:i] is a subsequence of s[0:j], what is the minimum index k where k > i, for which
        t[k:] is a subsequence of s[j+1:]
        (edge case, start and end of strings)

        abacaba
        bzaa
        0 -> j = 1
        1 -> not found

        3 -> j = 6
        2 -> j = 4
        1 -> not found
        0 -> not found
        """

        left = [None] * len(t)
        right = [None] * len(t)
        j = 0
        for i in range(0, len(t)):
            while j < len(s) and s[j] != t[i]:
                j+=1
            left[i] = j
            j+=1

        j = len(s) - 1
        for i in range(len(t) - 1, -1, -1):
            while j >= 0 and s[j] != t[i]:
                j-=1
            right[i] = j
            j -=1

        ans = len(t)   

        k = 0
        for i in range(0, len(t)):
            k = max(k, i+1)
            while k < len(right) and right[k] <= left[i]:
                k+=1
            if k < len(right) and right[k] > left[i]:
                ans = min(ans, (k-i-1)) 
            if left[i] < len(s):
                ans = min(ans, len(t)-i-1)
            if right[i] >= 0:
                ans = min(ans, i)
        return ans
25100