<p>There are <code>n</code> servers numbered from <code>0</code> to <code>n - 1</code> connected by undirected server-to-server <code>connections</code> forming a network where <code>connections[i] = [a<sub>i</sub>, b<sub>i</sub>]</code> represents a connection between servers <code>a<sub>i</sub></code> and <code>b<sub>i</sub></code>. Any server can reach other servers directly or indirectly through the network.</p>

<p>A <em>critical connection</em> is a connection that, if removed, will make some servers unable to reach some other server.</p>

<p>Return all critical connections in the network in any order.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>
<img alt="" src="https://assets.leetcode.com/uploads/2019/09/03/1537_ex1_2.png" style="width: 198px; height: 248px;" />
<pre>
<strong>Input:</strong> n = 4, connections = [[0,1],[1,2],[2,0],[1,3]]
<strong>Output:</strong> [[1,3]]
<strong>Explanation:</strong> [[3,1]] is also accepted.
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> n = 2, connections = [[0,1]]
<strong>Output:</strong> [[0,1]]
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>2 &lt;= n &lt;= 10<sup>5</sup></code></li>
	<li><code>n - 1 &lt;= connections.length &lt;= 10<sup>5</sup></code></li>
	<li><code>0 &lt;= a<sub>i</sub>, b<sub>i</sub> &lt;= n - 1</code></li>
	<li><code>a<sub>i</sub> != b<sub>i</sub></code></li>
	<li>There are no repeated connections.</li>
</ul>

from collections import defaultdict
from math import inf
from typing import List

def adjacency_list_from_edge_list(edge_list):
    adjacency_list = defaultdict(list)
    for u, v in edge_list:
        adjacency_list[u].append(v)
        adjacency_list[v].append(u)
    return adjacency_list

def cut_edges(adjacency_list, *, root):
    vertices = adjacency_list.keys()
    visited = {v: False for v in vertices}
    dfsnum = {v: inf for v in vertices}
    lownum = {v: inf for v in vertices}
    parent_of = {v: None for v in vertices}
    cut_edges = []
    current_dfs_num = 0

    def neighbors(node):
        yield from adjacency_list[node]

    def mark_visited(node):
        visited[node] = True

    def is_unvisited(node):
        return not visited[node]

    def is_back_edge(edge):
        u, v = edge
        return v != parent_of[u]

    def dfs(node):
        nonlocal current_dfs_num
        mark_visited(node)
        dfsnum[node] = lownum[node] = current_dfs_num
        current_dfs_num += 1

        for neighbor in neighbors(node):
            if is_unvisited(neighbor):
                parent_of[neighbor] = node
                dfs(neighbor)
                lownum[node] = min(lownum[node], lownum[neighbor])

                if lownum[neighbor] > dfsnum[node]:
                    cut_edges.append([node, neighbor])
            elif is_back_edge((node, neighbor)):
                lownum[node] = min(lownum[node], dfsnum[neighbor])

    for vertex in vertices:
        if is_unvisited(vertex):
            current_dfs_num = 0
            dfs(vertex)

    return cut_edges

class Solution:
    def criticalConnections(self, n: int, connections: List[List[int]]) -> List[List[int]]:
        adjacency_list = adjacency_list_from_edge_list(connections)
        return cut_edges(adjacency_list, root=0)


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