<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>

#include <vector>
#include <algorithm>

using namespace std;

class UnionFind {
public:
    vector<int> parent, rank;
    
    UnionFind(int n) {
        parent.resize(n);
        rank.resize(n, 1);
        for (int i = 0; i < n; ++i)
            parent[i] = i;
    }
    
    int find(int u) {
        if (parent[u] != u) {
            parent[u] = find(parent[u]);
        }
        return parent[u];
    }
    
    bool unionSets(int u, int v) {
        int rootU = find(u);
        int rootV = find(v);
        if (rootU != rootV) {
            if (rank[rootU] > rank[rootV]) {
                parent[rootV] = rootU;
            } else if (rank[rootU] < rank[rootV]) {
                parent[rootU] = rootV;
            } else {
                parent[rootV] = rootU;
                rank[rootU]++;
            }
            return true;
        }
        return false;
    }
};

class Solution {
public:
    int kruskal(int n, vector<vector<int>>& edges, int excludeEdge = -1, vector<int>* includeEdge = nullptr) {
        UnionFind uf(n);
        int weight = 0, edgesUsed = 0;

        if (includeEdge) {
            uf.unionSets((*includeEdge)[0], (*includeEdge)[1]);
            weight += (*includeEdge)[2];
            edgesUsed++;
        }
        
        for (int i = 0; i < edges.size(); ++i) {
            if (i == excludeEdge) continue;
            if (uf.unionSets(edges[i][0], edges[i][1])) {
                weight += edges[i][2];
                edgesUsed++;
            }
            if (edgesUsed == n - 1) break;
        }

        return (edgesUsed == n - 1) ? weight : INT_MAX;
    }

    vector<vector<int>> findCriticalAndPseudoCriticalEdges(int n, vector<vector<int>>& edges) {
        int m = edges.size();
        vector<vector<int>> indexedEdges;
        for (int i = 0; i < m; ++i) {
            indexedEdges.push_back({edges[i][0], edges[i][1], edges[i][2], i});
        }
        
        sort(indexedEdges.begin(), indexedEdges.end(), [](const vector<int>& a, const vector<int>& b) {
            return a[2] < b[2];
        });

        int mstWeight = kruskal(n, indexedEdges);

        vector<int> criticalEdges, pseudoCriticalEdges;
        
        for (int i = 0; i < m; ++i) {
            // Check if it's a critical edge
            if (kruskal(n, indexedEdges, i) > mstWeight) {
                criticalEdges.push_back(indexedEdges[i][3]);
            } 
            // Check if it's a pseudo-critical edge
            else if (kruskal(n, indexedEdges, -1, &indexedEdges[i]) == mstWeight) {
                pseudoCriticalEdges.push_back(indexedEdges[i][3]);
            }
        }

        return {criticalEdges, pseudoCriticalEdges};
    }
};


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