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Critical Connections in a Network

Certification: Intermediate Level Accuracy: 0% Submissions: 1 Points: 10

You are given an undirected connected graph with n nodes labeled from 0 to n-1 and a list of edges where edges[i] = [u, v] represents 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: 
      
      
    • Step 1: Analyze the given graph with 4 nodes and 4 edges.
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    • Step 2: Identify that nodes 0, 1, and 2 form a cycle.
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    • Step 3: Edge [1,3] is the only critical connection because if removed, node 3 will be isolated.
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    • Step 4: Return [[1,3]] as the result.
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Example 2
    
    
  • Input: n = 6, connections = [[0,1],[1,2],[2,0],[1,3],[3,4],[4,5],[5,3]]
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  • Output: []
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  • Explanation: 
      
      
    • Step 1: Analyze the given graph with 6 nodes and 7 edges.
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    • Step 2: Identify that nodes 0, 1, and 2 form one cycle.
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    • Step 3: Nodes 3, 4, and 5 form another cycle.
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    • Step 4: Every edge in the graph is part of a cycle, so removing any single edge will not disconnect the graph.
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    • Step 5: Return an empty list as there are no critical connections.
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Constraints
    
    
  • 1 ≤ n ≤ 10^5
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  • n-1 ≤ connections.length ≤ 10^5
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  • 0 ≤ connections[i][0], connections[i][1] ≤ n-1
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  • connections[i][0] != connections[i][1]
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  • There are no repeated connections.
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  • Time Complexity: O(V+E) where V is the number of vertices and E is the number of edges
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  • Space Complexity: O(V+E)
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GraphAlgorithmsIBMLTIMindtree
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Hint
Solution

Solution Hints

    
  
  • Use Tarjan's algorithm to find bridges in the graph.
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  • Keep track of discovery time and low time for each node.
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  • A bridge exists if the low value of the adjacent node is greater than the discovery time of the current node.
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  • Build an adjacency list representation of the graph for efficient traversal.
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  • Use depth-first search (DFS) to identify critical connections.
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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, lowest reachable time, and visited status.

 Step 3: Implement a DFS function that updates these values as we traverse the graph.

 Step 4: For each node, check if any of its edges is a bridge by comparing discovery and low times.

 Step 5: A bridge exists if the low value of the adjacent node is strictly greater than the discovery time of the current node.

 Step 6: Collect all bridges found during the traversal.

 Step 7: Return the list of bridges (critical connections).

Submitted Code :