<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 {
public:
    void dfs(int node, int parent, unordered_map<int,bool>& visited, vector<int>& disc, vector<int>& low, vector<vector<int>>& adj, vector<vector<int>>& res, int& timer) {
        visited[node] = true;
        disc[node] = low[node] = timer++;
        
        // Traverse all neighbors of the current node
        for (auto it : adj[node]) {
            if (it == parent) {
                continue; // If it's the parent node, ignore it
            }
            
            if (!visited[it]) { // If not visited, apply DFS
                dfs(it, node, visited, disc, low, adj, res, timer);
                
                // After recursion, update the low-link value of the node
                low[node] = min(low[node], low[it]);
                
                // Check if the edge (node, it) is a bridge
                if (low[it] > disc[node]) {
                    res.push_back({node, it});
                }
            } else {
                // If already visited, update the low-link value
                low[node] = min(low[node], disc[it]);
            }
        }
    }
    
    vector<vector<int>> criticalConnections(int n, vector<vector<int>>& connections) {
        vector<vector<int>> adj(n); // Adjacency list representation
        vector<vector<int>> res; // Store the result (critical connections)
        
        // Build the graph from connections
        for (auto& conn : connections) {
            adj[conn[0]].push_back(conn[1]);
            adj[conn[1]].push_back(conn[0]);
        }
        
        // Declare auxiliary data structures
        vector<int> disc(n, -1); // Discovery times of visited vertices
        vector<int> low(n, -1);  // Lowest discovery time reachable
        unordered_map<int, bool> visited; // Visited array
        int timer = 0;
        
        // Run DFS from node 0 (assuming the graph is connected)
        for (int i = 0; i < n; ++i) {
            if (!visited[i]) {
                dfs(i, -1, visited, disc, low, adj, res, timer);
            }
        }
        
        return res; // Return the critical connections
    }
};

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