<p>A <strong>transformation sequence</strong> from word <code>beginWord</code> to word <code>endWord</code> using a dictionary <code>wordList</code> is a sequence of words <code>beginWord -&gt; s<sub>1</sub> -&gt; s<sub>2</sub> -&gt; ... -&gt; s<sub>k</sub></code> such that:</p>

<ul>
	<li>Every adjacent pair of words differs by a single letter.</li>
	<li>Every <code>s<sub>i</sub></code> for <code>1 &lt;= i &lt;= k</code> is in <code>wordList</code>. Note that <code>beginWord</code> does not need to be in <code>wordList</code>.</li>
	<li><code>s<sub>k</sub> == endWord</code></li>
</ul>

<p>Given two words, <code>beginWord</code> and <code>endWord</code>, and a dictionary <code>wordList</code>, return <em>all the <strong>shortest transformation sequences</strong> from</em> <code>beginWord</code> <em>to</em> <code>endWord</code><em>, or an empty list if no such sequence exists. Each sequence should be returned as a list of the words </em><code>[beginWord, s<sub>1</sub>, s<sub>2</sub>, ..., s<sub>k</sub>]</code>.</p>

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

<pre>
<strong>Input:</strong> beginWord = &quot;hit&quot;, endWord = &quot;cog&quot;, wordList = [&quot;hot&quot;,&quot;dot&quot;,&quot;dog&quot;,&quot;lot&quot;,&quot;log&quot;,&quot;cog&quot;]
<strong>Output:</strong> [[&quot;hit&quot;,&quot;hot&quot;,&quot;dot&quot;,&quot;dog&quot;,&quot;cog&quot;],[&quot;hit&quot;,&quot;hot&quot;,&quot;lot&quot;,&quot;log&quot;,&quot;cog&quot;]]
<strong>Explanation:</strong>&nbsp;There are 2 shortest transformation sequences:
&quot;hit&quot; -&gt; &quot;hot&quot; -&gt; &quot;dot&quot; -&gt; &quot;dog&quot; -&gt; &quot;cog&quot;
&quot;hit&quot; -&gt; &quot;hot&quot; -&gt; &quot;lot&quot; -&gt; &quot;log&quot; -&gt; &quot;cog&quot;
</pre>

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

<pre>
<strong>Input:</strong> beginWord = &quot;hit&quot;, endWord = &quot;cog&quot;, wordList = [&quot;hot&quot;,&quot;dot&quot;,&quot;dog&quot;,&quot;lot&quot;,&quot;log&quot;]
<strong>Output:</strong> []
<strong>Explanation:</strong> The endWord &quot;cog&quot; is not in wordList, therefore there is no valid transformation sequence.
</pre>

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

<ul>
	<li><code>1 &lt;= beginWord.length &lt;= 5</code></li>
	<li><code>endWord.length == beginWord.length</code></li>
	<li><code>1 &lt;= wordList.length &lt;= 500</code></li>
	<li><code>wordList[i].length == beginWord.length</code></li>
	<li><code>beginWord</code>, <code>endWord</code>, and <code>wordList[i]</code> consist of lowercase English letters.</li>
	<li><code>beginWord != endWord</code></li>
	<li>All the words in <code>wordList</code> are <strong>unique</strong>.</li>
	<li>The <strong>sum</strong> of all shortest transformation sequences does not exceed <code>10<sup>5</sup></code>.</li>
</ul>

# in: str beginWord, str endWord, wordList
# out: shortest transformation sequence between begin and end word

# rules: we can only change one letter at a time
# beginWord not necessarily in wordList
# sequence may not exist

# we can try moving in a graph like structure
# hit -> *it, h*t, hi*
# each option maps to certain word possibilities
# we can map out all these possibilities as a graph
# will take O(Letters*words) to construct
# then we can build up best paths from root to each word using BFS



# eg: Input: beginWord = "hit", endWord = "cog", wordList = ["hot","dot","dog","lot","log","cog"]
# Output: [["hit","hot","dot","dog","cog"],["hit","hot","lot","log","cog"]]

# T: O(letters * words * ?)
# S: O(letters * words)

from collections import defaultdict
from collections import deque

class Solution:
    def findLadders(self, beginWord: str, endWord: str, wordList: List[str]) -> List[List[str]]:
        word_graph = defaultdict(set)

        # create map of all related words
        for word in wordList:
            for i in range(len(word)):
                key = word[:i] + "*" + word[i + 1:]
                word_graph[key].add(word)

        # Queue for BFS to find best paths
        q = deque([(beginWord, 0, [beginWord])])
        # closest words to each word
        closest_words = defaultdict(set)
        # min distances to start word
        distances = dict()
        distances[beginWord] = 0

        result = []

        while q:
            word, dist, path = q.popleft()
            for i in range(len(word)):
                key = word[:i] + "*" + word[i + 1:]
                for next_word in word_graph[key]:
                    # if we have not visited the word, or we are at a min distance to this word
                    if next_word not in distances or distances[next_word] >= dist + 1:
                        # add word to closest_words since it is on a short path from beginWord
                        closest_words[next_word].add(word)
                        # visit the next word we haven't yet seen
                        if next_word not in distances:
                            distances[next_word] = dist + 1
                            q.append((next_word, dist + 1, path + [next_word]))

        # go end to beginning along best path
        def collect_paths(word, path):
            good_paths = []
            if word == beginWord:
                return [path]
            else:
                for next_word in closest_words[word]:
                    good_paths.extend(collect_paths(next_word, [next_word] + path))
            return good_paths

        return collect_paths(endWord, [endWord])
17700