<p>Given an array of strings <code>words</code>, return <em>the smallest string that contains each string in</em> <code>words</code> <em>as a substring</em>. If there are multiple valid strings of the smallest length, return <strong>any of them</strong>.</p>

<p>You may assume that no string in <code>words</code> is a substring of another string in <code>words</code>.</p>

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

<pre>
<strong>Input:</strong> words = [&quot;alex&quot;,&quot;loves&quot;,&quot;leetcode&quot;]
<strong>Output:</strong> &quot;alexlovesleetcode&quot;
<strong>Explanation:</strong> All permutations of &quot;alex&quot;,&quot;loves&quot;,&quot;leetcode&quot; would also be accepted.
</pre>

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

<pre>
<strong>Input:</strong> words = [&quot;catg&quot;,&quot;ctaagt&quot;,&quot;gcta&quot;,&quot;ttca&quot;,&quot;atgcatc&quot;]
<strong>Output:</strong> &quot;gctaagttcatgcatc&quot;
</pre>

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

<ul>
	<li><code>1 &lt;= words.length &lt;= 12</code></li>
	<li><code>1 &lt;= words[i].length &lt;= 20</code></li>
	<li><code>words[i]</code> consists of lowercase English letters.</li>
	<li>All the strings of <code>words</code> are <strong>unique</strong>.</li>
</ul>

class Solution:
        
    def shortestSuperstring(self, words):
        n = len(words)
    
        # Precompute overlaps
        overlaps = [[0] * n for _ in range(n)]
        for i in range(n):
            for j in range(n):
                if i != j:
                    # max overlap of suffix of words[i] with prefix of words[j]
                    for k in range(min(len(words[i]), len(words[j])), 0, -1):
                        if words[i][-k:] == words[j][:k]:
                            overlaps[i][j] = k
                            break
        
        # DP with memoization
        @lru_cache(None)
        def dp(mask, last):
            if mask == (1 << n) - 1:
                return 0  # all strings are included
            
            res = float('inf')
            for i in range(n):
                if mask & (1 << i) == 0:  # if i-th string is not used yet
                    overlap = overlaps[last][i] if last != -1 else 0
                    res = min(res, dp(mask | (1 << i), i) + len(words[i]) - overlap)
            return res
        
        # Reconstruct the path
        def build_path():
            mask = 0
            last = -1
            result = ""
            for _ in range(n):
                best = float('inf')
                best_i = -1
                for i in range(n):
                    if mask & (1 << i) == 0:  # if i-th string is not used yet
                        overlap = overlaps[last][i] if last != -1 else 0
                        candidate = dp(mask | (1 << i), i) + len(words[i]) - overlap
                        if candidate < best:
                            best = candidate
                            best_i = i
                if last == -1:
                    result += words[best_i]
                else:
                    result += words[best_i][overlaps[last][best_i]:]
                mask |= (1 << best_i)
                last = best_i
            
            return result
        
        return build_path()

        

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