<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: List[str]) -> str:
        n = len(words)
        save = [[0]*n for i in range(n)]
        for i in range(n):
            w = words[i]
            for j in range(n):
                if i == j:
                    continue
                k = min(len(w), len(words[j]))
                while k:
                    if w[-k:] == words[j][:k]:
                        break
                    k -= 1
                save[i][j] = k
        
        d = [[-1]*n for i in range((1<<n)+1)]
        added = [[0]*n for i in range((1<<n)+1)]
        def dfs(mask: int, last: int) -> int:
            if mask == (1<<n)-1:
                return 0
            if d[mask][last] != -1:
                return d[mask][last]

            maxres, maxi = -1, -1
            for i in range(n):
                if mask & (1<<i):
                    continue
                res = dfs(mask|(1<<i), i)
                if mask:
                    res += save[last][i]
                if maxi == -1 or res > maxres:
                    maxres = res
                    maxi = i
            
            d[mask][last] = maxres
            added[mask][last] = maxi
            return maxres
        
        mask, last, ans = 0, 0, []
        dfs(mask, 0)
        while mask != (1<<n)-1:
            nxt = added[mask][last]
            if mask:
                ans.append(words[nxt][save[last][nxt]:])
            else:
                ans.append(words[nxt])
            mask |= 1<<nxt
            last = nxt
        
        return ''.join(ans)
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