<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:
        """
        dp[i][j] = min(dp[i-1][k] + d(k, j))
        """
        @cache
        def dist(k, j):
            len_k = len(words[k])
            len_j = len(words[j])
            r = min(len_k, len_j)
            for overlap_len in range(r, -1, -1):
                if words[k][len_k - overlap_len:len_k] == words[j][:overlap_len]:
                    return len_j - overlap_len
            return len_j
        limit = 1 << len(words)
        dp = [["" for i in range(len(words))] for j in range(limit)]

        for i in range(limit):
            for j in range(len(words)):
                if i == (1 << j):
                    dp[i][j] = words[j]
                elif i & (1 << j):
                    mask = i - (1 << j)
                    for k in range(len(words)):
                        if mask & (1 << k):
                            d = dist(k, j)
                            new_str = dp[mask][k] + words[j][len(words[j]) - d:len(words[j])]
                            if dp[i][j] == "" or len(new_str) < len(dp[i][j]) or (len(new_str) == len(dp[i][j]) and new_str < dp[i][j]):
                                dp[i][j] = new_str
        ret = ""
        #print(dp[-1][:])
        for cur_str in dp[-1][:]:
            if ret == "" or len(cur_str) < len(ret) or (len(cur_str) == len(ret) and cur_str < ret):
                ret = cur_str
        return ret

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