<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 UnionFind:
    def __init__(self, n):
        self.parent = list(range(n))
        self.rank = [0] * n
    def find(self, x):
        if self.parent[x] != x:
            self.parent[x] = self.find(self.parent[x])
        return self.parent[x]
    def union(self, x, y):
        rootx, rooty = self.find(x), self.find(y)
        if rootx == rooty: return False
        if self.rank[rootx] > self.rank[rooty]:
            self.parent[rooty] = rootx
        else:
            self.parent[rootx] = rooty
            if self.rank[rootx] == self.rank[rooty]:
                self.rank[rooty] += 1
        return True
    
class Solution:
    def findCriticalAndPseudoCriticalEdges(self, n: int, edges: List[List[int]]) -> List[List[int]]:
        def dfs(node, depth, ancestor):
            levels[node] = depth
            for neighbor, edge_index in adjacency_list[node]:
                if neighbor == ancestor:
                    continue
                if levels[neighbor] == -1:
                    levels[node] = min(levels[node], dfs(neighbor, depth + 1, node))
                else:
                    levels[node] = min(levels[node], levels[neighbor])
                if levels[neighbor] >= depth + 1 and edge_index not in pseudo_critical:
                    critical.add(edge_index)
            return levels[node]

        critical, pseudo_critical = set(), set()

        weight_to_edges = {w: [] for w in {edge[2] for edge in edges}}
        for i, (u, v, w) in enumerate(edges):
            weight_to_edges[w].append([u, v, i])
        
        union_find = UnionFind(n)
        
        for weight in sorted(weight_to_edges):
            edge_lookup = defaultdict(set)
            
            for u, v, edge_index in weight_to_edges[weight]:
                rootU, rootV = union_find.find(u), union_find.find(v)
                
                if rootU != rootV:
                    union_pair = tuple(sorted((rootU, rootV)))
                    edge_lookup[union_pair].add(edge_index)
            
            mst_edges, adjacency_list = [], defaultdict(list)
            
            for (rootU, rootV), edge_indexes in edge_lookup.items():
                if len(edge_indexes) > 1:
                    pseudo_critical.update(edge_indexes)
                
                edge_idx = edge_indexes.pop()
                adjacency_list[rootU].append((rootV, edge_idx))
                adjacency_list[rootV].append((rootU, edge_idx))
                mst_edges.append((rootU, rootV, edge_idx))
                union_find.union(rootU, rootV)
            
            levels = [-1] * n
            
            for u, v, _ in mst_edges:
                if levels[u] == -1:
                    dfs(u, 0, -1)
            
            for _, _, edge_index in mst_edges:
                if edge_index not in critical:
                    pseudo_critical.add(edge_index)

        return [list(critical), list(pseudo_critical)]
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