<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 math import inf

def adjacency_list_from_edge_list(edge_list):
    n = len(edge_list)
    adjacency_list = [[] for i in range(n + 1)] 

    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):
    n = len(adjacency_list)
    vertices = [i for i in range(n)]
    visited = {v : False for v in vertices}
    dfs_num = {v : inf for v in vertices}
    low_num = {v : inf for v in vertices}
    parent_of = {v : None for v in vertices}
    counter = 0
    cut_edges = []

    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 depth_first_search(root, counter):
        mark_visited(root)
        dfs_num[root] = counter
        low_num[root] = counter
        counter += 1

        for neighbor in neighbors(root):
            parent = root
            child = neighbor
            if is_unvisited(neighbor):
                parent_of[child] = parent
                depth_first_search(child, counter)
                if low_num[child] < low_num[parent]:
                    low_num[parent] = low_num[child]
                if low_num[child] > dfs_num[parent]:
                    cut_edges.append([parent, child])
            elif is_back_edge([parent, child]):
                low_num[root] = min(low_num[root], dfs_num[neighbor])
                
    depth_first_search(root, counter)
    return cut_edges


class Solution:
    def criticalConnections(self, n, connections):
        adjacency_list = adjacency_list_from_edge_list(connections)
        return cut_edges(adjacency_list, 0)
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