<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>

def cut_edges(adjacency_list):
        VERTICES = adjacency_list.keys()
        VISITED = {v:False for v in VERTICES}
        DFSNUM = {v:0 for v in VERTICES}
        LOWNUM = {v:math.inf for v in VERTICES}
        PARENT_OF = {v:None for v in VERTICES}
        root = 0
        DFSNUM[root] = 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 dfs(root):
            mark_visited(root)
            LOWNUM[root] = min(LOWNUM[root], DFSNUM[root])
            for neighbor in neighbors(root):
                edge = (root, neighbor)
                if is_unvisited(neighbor):
                    parent, child = root, neighbor
                    PARENT_OF[child] = parent
                    DFSNUM[child] = 1 + DFSNUM[parent]
                    dfs(child)
                    if LOWNUM[child] > DFSNUM[parent]:
                        CUT_EDGES.append(edge)
                    LOWNUM[parent] = min(LOWNUM[parent], LOWNUM[child])
                elif is_back_edge(edge):
                    LOWNUM[root] = min(LOWNUM[root], DFSNUM[neighbor])

        for vertex in VERTICES:
            if is_unvisited(vertex):
                dfs(vertex)
        return CUT_EDGES

class Solution:
    def criticalConnections(self, n: int, connections: List[List[int]]) -> List[List[int]]:
        adjacency_list = collections.defaultdict(list)
        for u, v in connections:
            adjacency_list[u].append(v)
            adjacency_list[v].append(u)
        return cut_edges(adjacency_list)
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