<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:
    // https://www.youtube.com/watch?v=83JnUxrLKJU

    int findParent(int a, vector<int>& parent) {
        if (parent[a] == -1) {
            return a;
        }
        return findParent(parent[a], parent);
    }

    bool doUnion(int a, int b, vector<int>& rank, vector<int>& parent) {
        int parentA = findParent(a, parent);
        int parentB = findParent(b, parent);
        if (parentA == parentB) {
            return false;
        }
        int rankParentA = rank[parentA];
        int rankParentB = rank[parentB];

        if (rankParentA > rankParentB) {
            parent[parentB] = parentA;
        } else if (rankParentA == rankParentB) {
            parent[parentB] = parentA;
            rank[parentA]++;
        } else {
            parent[parentA] = parentB;
        }
        return true;
    }

    pair<int, int> findMstWeightWithKruskals(vector<vector<int>>& edges, int n,
                                             vector<int>& mustIncludeEdge) {
        // though this approach makes use of heap, we can achieve similar
        // functionality with sorting since we only pop from heap and dont push
        // anything
        vector<int> rank(n, 0);
        vector<int> parent(n, -1);
        if (mustIncludeEdge[0] != -1 && mustIncludeEdge[1] != -1) {
            doUnion(mustIncludeEdge[0], mustIncludeEdge[1], rank, parent);
        }
        int mstWeight = mustIncludeEdge[2] == -1 ? 0 : mustIncludeEdge[2];
        int edgesAdded = mustIncludeEdge[2] == -1 ? 0 : 1;
        for (auto itr : edges) {
            int node1 = itr[0];
            int node2 = itr[1];
            int weight = itr[2];
            if (node1 == -1 && node2 == -1) {
                continue;
            }
            if (doUnion(node1, node2, rank, parent)) {
                mstWeight += weight;
                edgesAdded++;
            }
        }
        return pair<int, int>{mstWeight, edgesAdded};
    }

    static bool comparator(const vector<int>& a, const vector<int>& b) {
        return a[2] < b[2];
    }

    vector<vector<int>>
    findCriticalAndPseudoCriticalEdges(int n, vector<vector<int>>& edges) {

        for (int i = 0; i < edges.size(); i++) {
            edges[i].push_back(i);
        }
        sort(edges.begin(), edges.end(), comparator);
        vector<int> mustIncludeNormal{-1, -1, -1};
        int mstWeightOriginal =
            findMstWeightWithKruskals(edges, n, mustIncludeNormal).first;
        vector<int> criticalEdgesIndex;
        vector<int> pseudoCriticalEdgesIndex;
        for (int i = 0; i < edges.size(); i++) {
            vector<int> curr = edges[i];
            int originalIndex = curr[3];
            vector<int> mustIncludeNormal{-1, -1, -1};
            edges[i][0] = -1;
            edges[i][1] = -1;
            pair<int, int> withEdgeRemoved =
                findMstWeightWithKruskals(edges, n, mustIncludeNormal);
            int currentMstWeightWithEdgeRemoved = withEdgeRemoved.first;
            int totalEdgesInMst = withEdgeRemoved.second;
            if (totalEdgesInMst < n - 1 ||
                currentMstWeightWithEdgeRemoved > mstWeightOriginal) {
                criticalEdgesIndex.push_back(originalIndex);
                edges[i] = curr;
                continue;
            }
            edges[i] = curr;

            pair<int, int> withEdgeForced =
                findMstWeightWithKruskals(edges, n, curr);
            int currentMstWeightWithEdgeAdded = withEdgeForced.first;

            if (currentMstWeightWithEdgeAdded == mstWeightOriginal) {
                pseudoCriticalEdgesIndex.push_back(originalIndex);
            }
        }
        vector<vector<int>> answer = {criticalEdgesIndex,
                                      pseudoCriticalEdgesIndex};

        return answer;
    }
};
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