<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:

            # Problme-d: Critical Connections
            # Problme-d: Critical Connections
        # Problme-d: Critical Connections
    # Problme-d: Critical Connections
    # Problem - d: Critical Connections: https://leetcode.com/problems/critical-connections-in-a-network/description/
    def criticalConnections(self, n: int, connections: list[list[int]]) -> list[list[int]]:

        def dfs(parent, node, adjacencyList, visited, minimumDepth, depth, criticalConnections):   
            visited[node] = depth
            minimumDepth[node] = depth

            for connectedServer in adjacencyList[node]:
                # if connectedServer and parent are same, then no need to traverse again as it is already done
                if connectedServer == parent:
                    continue
                # if not visited yet, call the dfs from this node (server)
                if connectedServer not in visited:
                    dfs(node, connectedServer, adjacencyList, visited, minimumDepth, depth+1, criticalConnections)

                # if minimumDepth of current connectedServer is bigger than the current depth, that means there is a critical connection.
                # Thus, add it to criticalConnections list.
                if minimumDepth[connectedServer] > depth:
                    criticalConnections.append([node, connectedServer])
                
                # update the minimumDepth of the current node
                minimumDepth[node] = min(minimumDepth[node], minimumDepth[connectedServer])        

        # creating an adjacency list based on the connections b/w servers (u, v)
        adjacencyList = defaultdict(set)
        for u, v in connections:
            adjacencyList[u].add(v)
            adjacencyList[v].add(u)

        # initialize the critical connections as an empty list
        criticalConnection = []
        # parent is -1 in start, as current there is no parent of node 0
        dfs(-1, 0, adjacencyList, {}, {}, 0, criticalConnection)
        return criticalConnection
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