<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 <iostream>
#include <vector>
#include <algorithm>
#include <climits>
#include <numeric>

class Solution {
public:
    int vertices, components;
    std::vector<int> parent;

    int find(int node) {
        while (true) {
            if (parent[node] < 0) return node;
            node = parent[node];
        }
    }

    void unionNodes(int node1, int node2) {
        int root1 = find(node1);
        int root2 = find(node2);
        if (root1 == root2) return;
        if (std::abs(parent[root1]) > std::abs(parent[root2])) {
            parent[root1] += parent[root2];
            parent[root2] = root1;
        } else {
            parent[root2] += parent[root1];
            parent[root1] = root2;
        }
        components--;
    }

    int calculateMST(const std::vector<std::vector<int>>& edges, int includeEdge, int skipEdge) {
        parent.assign(vertices, -1);
        components = vertices;
        int totalWeight = 0;

        if (includeEdge != -1) {
            totalWeight += edges[includeEdge][0];
            unionNodes(find(edges[includeEdge][1]), find(edges[includeEdge][2]));
        }

        for (int i = 0; i < edges.size(); ++i) {
            if (i == skipEdge) continue;
            int weight = edges[i][0];
            int u = edges[i][1];
            int v = edges[i][2];
            int root1 = find(u);
            int root2 = find(v);

            if (root1 != root2) {
                totalWeight += weight;
                unionNodes(root1, root2);
            }
        }

        return components != 1 ? INT_MAX : totalWeight;
    }

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

        this->vertices = n;
        int minMSTWeight = calculateMST(edgeList, -1, -1);

        for (int i = 0; i < edgeList.size(); ++i) {
            int skippedMSTWeight = calculateMST(edgeList, -1, i);
            int includedMSTWeight = calculateMST(edgeList, i, -1);

            if (skippedMSTWeight > minMSTWeight) {
                result[0].push_back(edgeList[i][3]);
            } else if (includedMSTWeight == minMSTWeight) {
                result[1].push_back(edgeList[i][3]);
            }
        }

        return result;
    }
};


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