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Critical Connections in a Network

Certification: Advanced Level Accuracy: 0% Submissions: 0 Points: 15

Write a C++ program to find all critical connections in a network. A critical connection is an edge that, if removed, will make some nodes unable to reach each other. Given a network with n nodes labeled from 0 to n-1 and a list of connections (edges) where each connection is represented by a pair [u, v] indicating that there is an undirected edge between nodes u and v, return all critical connections in the network in any order.

Example 1
    
  
  • Input: n = 4, connections = [[0,1],[1,2],[2,0],[1,3]]
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  • Output: [[1,3]]
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  • Explanation: 
      
      
    • The edge [1,3] is the only critical connection in the network.
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    • If this edge is removed, node 3 will be disconnected from the rest of the network.
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Example 2
    
  
  • Input: n = 5, connections = [[0,1],[1,2],[2,0],[1,3],[3,4]]
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  • Output: [[1,3],[3,4]]
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  • Explanation: 
      
      
    • The edges [1,3] and [3,4] are critical connections.
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    • If either is removed, the network will have disconnected components.
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Constraints
    
  
  • 1 ≤ n ≤ 10^5
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  • 1 ≤ connections.length ≤ 10^5
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  • connections[i][0] != connections[i][1]
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  • No repeated connections
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  • Time Complexity: O(V + E) where V is number of vertices and E is number of edges
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  • Space Complexity: O(V + E)
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AlgorithmsEYPhillips
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Hint
Solution

Solution Hints

    
  
  • Use Tarjan's algorithm to find bridges in the graph
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  • Implement a DFS traversal to discover and classify edges
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  • Keep track of discovery times and lowest reachable vertex
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  • An edge is a bridge (critical connection) if the lowest reachable vertex from v is greater than the discovery time of u
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  • Return all identified bridges as the answer
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Steps to solve by this approach:

 Step 1: Build an adjacency list representation of the graph.
 Step 2: Initialize arrays to track discovery time and lowest reachable node for each vertex.
 Step 3: Perform a depth-first search (DFS) traversal of the graph.
 Step 4: For each vertex, track the discovery time and update the lowest reachable node.
 Step 5: During DFS, when we visit a node's neighbor, we update the lowest reachable node.
 Step 6: If the lowest reachable from a neighbor is greater than the discovery time of the current node, the edge is a bridge.
 Step 7: Collect and return all bridge edges identified during the traversal.

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