<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>

class Solution {
public:
    vector<vector<string>> findLadders(string beginWord, string endWord, vector<string>& wordList) {
        set<string> words;
        words.insert(beginWord);
        for (string word : wordList) {
            words.insert(word);
        }
        vector<string> ids;
        ids.insert(ids.begin(), words.begin(), words.end());
        int n = ids.size();
        int s = distance(ids.begin(), lower_bound(ids.begin(), ids.end(), beginWord));
        vector<map<string,vector<int>>> edgeProp(beginWord.size());
        vector<vector<int>> edges(ids.size());
        for (int i = 0; i < ids.size(); i++) {
            string word = ids[i];
            for (int j = 0; j < word.size(); j++) {
                string sub = word.substr(0,j) + word.substr(j+1);
                for (int v : edgeProp[j][sub]) {
                    edges[i].push_back(v);
                    edges[v].push_back(i);
                }
                edgeProp[j][sub].push_back(i);
            }
        }
        typedef pair<int, int> P;
        vector<int> dist(n, INT_MAX);
        vector<vector<int>> parent(n);
        priority_queue<P, vector<P>, greater<P>> pq;
        pq.push({0, s});
        dist[s] = 0;
        while (pq.size()) {
            auto [d, x] = pq.top(); pq.pop();
            if (d == dist[x]) {
                for (int v : edges[x]) {
                    if (d+1 == dist[v]) {
                        parent[v].push_back(x);
                    } else if (d + 1 < dist[v]) {
                        parent[v].clear();
                        parent[v].push_back(x);
                        dist[v] = d + 1;
                        pq.push({d+1, v});
                    }
                }
            }
        }
        vector<int> path;
        vector<vector<string>> res;
        function<void(int)> solve = [&](int x) {
            path.push_back(x);
            if (x == s) {
                vector<string> out;
                for (int i = path.size()-1; i >= 0; i--) {
                    out.push_back(ids[path[i]]);
                }
                res.push_back(out);
            } else {
                for (int v : parent[x]) {
                    solve(v);
                }
            }
            path.pop_back();
        };
        int e = distance(ids.begin(), lower_bound(ids.begin(), ids.end(), endWord)); 
        if (ids[e] == endWord) solve(e);
        return res;
    }
};
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