<p>Given a weighted undirected connected graph with <code>n</code>&nbsp;vertices numbered from <code>0</code> to <code>n - 1</code>,&nbsp;and an array <code>edges</code>&nbsp;where <code>edges[i] = [a<sub>i</sub>, b<sub>i</sub>, weight<sub>i</sub>]</code> represents a bidirectional and weighted edge between nodes&nbsp;<code>a<sub>i</sub></code>&nbsp;and <code>b<sub>i</sub></code>. A minimum spanning tree (MST) is a subset of the graph&#39;s edges that connects all vertices without cycles&nbsp;and with the minimum possible total edge weight.</p>

<p>Find <em>all the critical and pseudo-critical edges in the given graph&#39;s minimum spanning tree (MST)</em>. An MST edge whose deletion from the graph would cause the MST weight to increase is called a&nbsp;<em>critical edge</em>. On&nbsp;the other hand, a pseudo-critical edge is that which can appear in some MSTs but not all.</p>

<p>Note that you can return the indices of the edges in any order.</p>

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

<p><img alt="" src="https://assets.leetcode.com/uploads/2020/06/04/ex1.png" style="width: 259px; height: 262px;" /></p>

<pre>
<strong>Input:</strong> n = 5, edges = [[0,1,1],[1,2,1],[2,3,2],[0,3,2],[0,4,3],[3,4,3],[1,4,6]]
<strong>Output:</strong> [[0,1],[2,3,4,5]]
<strong>Explanation:</strong> The figure above describes the graph.
The following figure shows all the possible MSTs:
<img alt="" src="https://assets.leetcode.com/uploads/2020/06/04/msts.png" style="width: 540px; height: 553px;" />
Notice that the two edges 0 and 1 appear in all MSTs, therefore they are critical edges, so we return them in the first list of the output.
The edges 2, 3, 4, and 5 are only part of some MSTs, therefore they are considered pseudo-critical edges. We add them to the second list of the output.
</pre>

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

<p><img alt="" src="https://assets.leetcode.com/uploads/2020/06/04/ex2.png" style="width: 247px; height: 253px;" /></p>

<pre>
<strong>Input:</strong> n = 4, edges = [[0,1,1],[1,2,1],[2,3,1],[0,3,1]]
<strong>Output:</strong> [[],[0,1,2,3]]
<strong>Explanation:</strong> We can observe that since all 4 edges have equal weight, choosing any 3 edges from the given 4 will yield an MST. Therefore all 4 edges are pseudo-critical.
</pre>

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

<ul>
	<li><code>2 &lt;= n &lt;= 100</code></li>
	<li><code>1 &lt;= edges.length &lt;= min(200, n * (n - 1) / 2)</code></li>
	<li><code>edges[i].length == 3</code></li>
	<li><code>0 &lt;= a<sub>i</sub> &lt; b<sub>i</sub> &lt; n</code></li>
	<li><code>1 &lt;= weight<sub>i</sub>&nbsp;&lt;= 1000</code></li>
	<li>All pairs <code>(a<sub>i</sub>, b<sub>i</sub>)</code> are <strong>distinct</strong>.</li>
</ul>

class Solution {
public:
    int mst_cost(vector<vector<pair<int,int>>> &g,int a=-1,int b=-1,int w_ab=0){
        priority_queue<pair<int,int>,vector<pair<int,int>>,greater<pair<int,int>>> pq;
        int mst = 0;
        int num_edge_mst = 0;
        unordered_set<int> included;         
        if(a==-1 && b==-1){
            int min_nbr = -1,w=1e8;
            for(int i=0;i<g[0].size();i++){
                auto nbr = g[0][i];
                if(nbr.second < w){
                    w = nbr.second;
                    min_nbr = nbr.first;
                }
            }
            a = 0;
            b = min_nbr;
            w_ab = w;
        }    
        included.insert(a);
        included.insert(b);
        pq.push({-1,a});
        pq.push({-1,b});
        mst += w_ab;
        num_edge_mst += 1;  
        while(!pq.empty() && num_edge_mst<g.size()-1){      
            auto f = pq.top();                          
            pq.pop();                                           
            if(f.first > 0 && (included.find(f.second)==included.end())){
                mst += f.first;
                num_edge_mst += 1;
                included.insert(f.second);
            }                               
            for(auto nbr:g[f.second]){
                if(included.find(nbr.first) != included.end()) continue;
                pq.push({nbr.second,nbr.first});
            }                          
        }                           
        return mst;
    }
    vector<vector<int>> findCriticalAndPseudoCriticalEdges(int n, vector<vector<int>>& edges) {
        vector<vector<int>> ans;
        vector<vector<pair<int,int>>> g(n);
        for(int i=0;i<edges.size();i++){
            g[edges[i][0]].push_back({edges[i][1],edges[i][2]});
            g[edges[i][1]].push_back({edges[i][0],edges[i][2]});
        }
        int MST = mst_cost(g);  
        vector<int> critical;
        vector<int> p_critical;
        for(int i=0;i<edges.size();i++){
            auto e = edges[i];
            int a = e[0],b = e[1],w = e[2];
            for(int j=0;j<g[a].size();j++){
                if(g[a][j].first == b) g[a][j].second = 1001;
            }
            for(int j=0;j<g[b].size();j++){
                if(g[b][j].first == a) g[b][j].second = 1001;
            }                                       
            int mst_e = mst_cost(g);                          
            if(mst_e > MST){
                critical.push_back(i);
            }
            for(int j=0;j<g[a].size();j++){
                if(g[a][j].first == b) g[a][j].second = w;
            }
            for(int j=0;j<g[b].size();j++){
                if(g[b][j].first == a) g[b][j].second = w;
            }
            
            int mst_e1 = mst_cost(g,a,b,w);             
            if(mst_e1 == MST){
                if(mst_e <= MST)
                p_critical.push_back(i);
            }
        }
        ans.push_back(critical);
        ans.push_back(p_critical);
        return ans;
    }
};
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