<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>

from collections import defaultdict
class Solution:
    def findAllConcatenatedWordsInADict(self, words: List[str]) -> List[str]:
        groups = defaultdict(list)
        for w in words:
            groups[w[0]].append(w)

        def match(word: str, start: int, another_word: str)-> bool:
            Lw = len(word)
            La = len(another_word)
            if start + La > Lw:
                return False
            for i in range(La):
                if word[start + i] != another_word[i]:
                    return False
            return True

        def is_concatenated(word: str) -> bool:
            L = len(word)
            dp = [0] * L
            matched = {-1}
            while True:
                next_matched = set()
                for mid in matched:
                    c = word[mid + 1]
                    for aw in groups.get(c, []):
                        if aw == word:
                            continue
                        if match(word, mid + 1, aw):
                            next_matched.add(mid + len(aw))
                if len(next_matched) == 0:
                    return False
                if (L-1) in next_matched:
                    return True
                matched = next_matched

            return False
        answer = []
        for w in words:
            if is_concatenated(w):
                answer.append(w)
        return answer


        
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