<p>Given an array of strings <code>words</code> (<strong>without duplicates</strong>), return <em>all the <strong>concatenated words</strong> in the given list of</em> <code>words</code>.</p>

<p>A <strong>concatenated word</strong> is defined as a string that is comprised entirely of at least two shorter words (not necessarily distinct)&nbsp;in the given array.</p>

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

<pre>
<strong>Input:</strong> words = [&quot;cat&quot;,&quot;cats&quot;,&quot;catsdogcats&quot;,&quot;dog&quot;,&quot;dogcatsdog&quot;,&quot;hippopotamuses&quot;,&quot;rat&quot;,&quot;ratcatdogcat&quot;]
<strong>Output:</strong> [&quot;catsdogcats&quot;,&quot;dogcatsdog&quot;,&quot;ratcatdogcat&quot;]
<strong>Explanation:</strong> &quot;catsdogcats&quot; can be concatenated by &quot;cats&quot;, &quot;dog&quot; and &quot;cats&quot;; 
&quot;dogcatsdog&quot; can be concatenated by &quot;dog&quot;, &quot;cats&quot; and &quot;dog&quot;; 
&quot;ratcatdogcat&quot; can be concatenated by &quot;rat&quot;, &quot;cat&quot;, &quot;dog&quot; and &quot;cat&quot;.</pre>

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

<pre>
<strong>Input:</strong> words = [&quot;cat&quot;,&quot;dog&quot;,&quot;catdog&quot;]
<strong>Output:</strong> [&quot;catdog&quot;]
</pre>

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

<ul>
	<li><code>1 &lt;= words.length &lt;= 10<sup>4</sup></code></li>
	<li><code>1 &lt;= words[i].length &lt;= 30</code></li>
	<li><code>words[i]</code> consists of only lowercase English letters.</li>
	<li>All the strings of <code>words</code> are <strong>unique</strong>.</li>
	<li><code>1 &lt;= sum(words[i].length) &lt;= 10<sup>5</sup></code></li>
</ul>

class Trie:

    def __init__(self):
        self.children = [None] * 26
        self.end = False

    def insert(self, word):
        root = self 
        for c in word:
            shift = ord(c) - ord("a")
            if not root.children[shift]:
                root.children[shift] = Trie()
            root = root.children[shift]
        root.end = True
        return


class Solution:
    def findAllConcatenatedWordsInADict(self, words: List[str]) -> List[str]:
        words.sort(key=lambda x: len(x))
        trie = Trie()

        @cache
        def dfs(root, start, word, concat):
            if start == len(word):
                return concat > 1
            for i in range(start, len(word)):
                shift = ord(word[i]) - ord("a")
                if root.children[shift]:
                    root = root.children[shift]
                    if root.end and dfs(trie, i + 1, word, concat + 1):
                        return True
                else:
                    return False
            return False

        res = []
        for word in words:
            root = trie
            if dfs(root, 0, word, 0):
                res.append(word)
            trie.insert(word)
        return res


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