<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:

        # 1. build prefix and suffix
        #   prefix[i]: index of t which has its left part a subsequence of s[:i]
        #              t[: prefix[i]] is a subsequence of s[: i]
        #   suffix[i]: index of t which has its right part a subsequence of s[i:]
        #               t[suffix[i] :] is a subsequence of s[i :]
        #   prefix is left-to-right scanning, while suffix is right-to-left scanning
        
        prefix = [-1] * len(s)
        i = j = 0
        while i < len(s):
            if j < len(t) and s[i] == t[j]:
                prefix[i] = j
                j += 1
            elif i > 0:
                prefix[i] = prefix[i - 1]
            i += 1

        if prefix[-1] == len(t) - 1:    # t is subsequence of s
            return 0

        suffix = [len(t)] * len(s)
        i, j = len(s) - 1, len(t) - 1
        while i >= 0:
            if j >= 0 and s[i] == t[j]:
                suffix[i] = j
                j -= 1
            elif i < len(s) - 1:
                suffix[i] = suffix[i + 1]
            i -= 1

        # 2. scan s to find the longest subsequence based on prefix and suffix
        # since t is not a subsequence of s, so prefix[i] < suffix[i]

        length = float('Inf')

        for idx in range(len(s)):
            # check if s[idx] is the first match for s[prefix[idx]] and s[suffix[idx]]
            # if so, need to remove one more chars
            if (prefix[idx] >= 0) and (idx == 0 or prefix[idx] > prefix[idx - 1]):
                pre_match = True    # s[idx] is the one to match t[prefix[idx]] for the first time
            else:
                pre_match = False
            if (suffix[idx] < len(t)) and (idx == len(s) - 1 or suffix[idx] < suffix[idx + 1]):
                suf_match = True    # s[idx] is the one to match t[suffix[idx]] for the first time
            else:
                suf_match = False

            if  pre_match and suf_match:
                length = min(length, suffix[idx] - prefix[idx])
            else:
                length = min(length, suffix[idx] - prefix[idx] - 1)

        return length
22300