<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>

def make_subsequence_lookup(s, t):
    lookup = [-1]

    s_ptr = 0
    t_ptr = 0
    while t_ptr < len(t):
        if s_ptr == len(s):
            lookup.append(None)
            t_ptr += 1
        elif s[s_ptr] == t[t_ptr]:
            lookup.append(s_ptr)
            s_ptr += 1
            t_ptr += 1
        else:
            s_ptr += 1
    return lookup

class Solution:
    def minimumScore(self, s: str, t: str) -> int:
        
        left_lookup = make_subsequence_lookup(s, t)

        

        rev_s = "".join(reversed(s))
        rev_t = "".join(reversed(t))
        right_lookup = make_subsequence_lookup(rev_s, rev_t)
        right_lookup = [
            (len(s) - 1 - l) if l is not None else None
            for l in reversed(right_lookup)
        ]
        
        # s = abacaba
        # t = bzaa
        # [-1, 1, None, None, None]
        # [None None 4 6 7]

        start_idx = 0
        end_idx = 0 

        best = float('inf')
        while start_idx < len(left_lookup) and left_lookup[start_idx] is not None:
            end_idx = max(end_idx, start_idx)
            while (
                (end_idx < len(right_lookup)) and 
                ((right_lookup[end_idx] is None) or  
                (left_lookup[start_idx] >= right_lookup[end_idx]))
                ):
                end_idx += 1
            
            best = min(best, end_idx - start_idx)
            start_idx += 1

        return best


        


        
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