<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 DisjointSet {
    public:
    vector<int> size, parent;
    DisjointSet(int n) {
        size.resize(n+1, 1);
        parent.resize(n+1);
        for (int i = 0; i <= n; i++) {
            parent[i] = i;
        }
    }
    
    int findUParent(int u) {
        if (parent[u] == u)
            return u;
        return parent[u] = findUParent(parent[u]);
    }
    
    void unionBySize(int u, int v) {
        int pu = findUParent(u);
        int pv = findUParent(v);
        
        if (pu == pv) return;
        
        if (size[pu] < size[pv]) {
            parent[pu] = pv;
            size[pv] += size[pu];
        } else {
            parent[pv] = pu;
            size[pu] += size[pv];
        }
    }
};

class Solution {
public:
    int kruskal(int n, vector<vector<int>>& edges, int skip, int addEdge) {
        DisjointSet ds(n);

        int mstWt = 0;

        if (addEdge != -1) {
            ds.unionBySize(edges[addEdge][0], edges[addEdge][1]);
            mstWt += edges[addEdge][2];
        }

        for (int i = 0; i < edges.size(); i++) {
            auto edge = edges[i];

            if (skip == i) continue;

            int u = edge[0];
            int v = edge[1];
            int wt = edge[2];

            if (ds.findUParent(u) != ds.findUParent(v)) {
                ds.unionBySize(u, v);
                mstWt += wt;
            }
        }

        int numComponents = 0;
        for (int i = 0; i < n; i++) {
            if (ds.findUParent(i) == i) numComponents++;
        }

        if (numComponents > 1) return 1e9;
        
        return mstWt;
    }

    vector<vector<int>> findCriticalAndPseudoCriticalEdges(int n, vector<vector<int>>& edges) {
        for (int i = 0; i < edges.size(); i++) {
            edges[i].push_back(i);
        }

        auto lambda = [](vector<int>& v1, vector<int>& v2) {
            return v1[2] < v2[2];
        };
        sort(edges.begin(), edges.end(), lambda);

        int mstWt = kruskal(n, edges, -1, -1);
        vector<int> critical;
        vector<int> pseudoCritical;

        for (int i = 0; i < edges.size(); i++) {
            if (kruskal(n, edges, i, -1) > mstWt) { // skipping ith edge
                critical.push_back(edges[i][3]);
            } else if (kruskal(n, edges, -1, i) == mstWt) { // force add ith edge
                pseudoCritical.push_back(edges[i][3]);
            }
        }

        return { critical, pseudoCritical };
    }
};
91909