<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>

class Solution:
    def criticalConnections(self, n: int, connections: List[List[int]]) -> List[List[int]]:
        global graph, N, critical_conns, timer, arrivals
        N = n
        critical_conns = []
        graph = collections.defaultdict(set)
        arrivals = {}
        timer = 0

        for u,v in connections:
            graph[u].add(v)
            graph[v].add(u)
        
        dfs(0, None)
        return critical_conns


def dfs(u, parent):
    record_entry(u)
    loopback = arrival_time(u)
    for v in graph[u]:
        if v == parent:
            continue
        if is_visited(v):
            loopback = min(loopback, arrival_time(v))
            continue
        child_loopback = dfs(v, u)
        if child_loopback > arrival_time(u):
            #found a critical connection, there no alternate route to connect back to ancestor nodes
            critical_conns.append([u,v])
        loopback = min(loopback, child_loopback)
    
    return loopback


def record_entry(u):
    global timer
    arrivals[u] = timer
    timer += 1

def arrival_time(u):
    return arrivals.get(u) 

def is_visited(u):
    return arrivals.get(u) != None
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