Binary TreesBinary Trees36
  1. 1Preorder Traversal of a Binary Tree using Recursion
  2. 2Preorder Traversal of a Binary Tree using Iteration
  3. 3Inorder Traversal of a Binary Tree using Recursion
  4. 4Inorder Traversal of a Binary Tree using Iteration
  5. 5Postorder Traversal of a Binary Tree Using Recursion
  6. 6Postorder Traversal of a Binary Tree using Iteration
  7. 7Level Order Traversal of a Binary Tree using Recursion
  8. 8Level Order Traversal of a Binary Tree using Iteration
  9. 9Reverse Level Order Traversal of a Binary Tree using Iteration
  10. 10Reverse Level Order Traversal of a Binary Tree using Recursion
  11. 11Find Height of a Binary Tree
  12. 12Find Diameter of a Binary Tree
  13. 13Find Mirror of a Binary Tree
  14. 14Left View of a Binary Tree
  15. 15Right View of a Binary Tree
  16. 16Top View of a Binary Tree
  17. 17Bottom View of a Binary Tree
  18. 18Zigzag Traversal of a Binary Tree
  19. 19Check if a Binary Tree is Balanced
  20. 20Diagonal Traversal of a Binary Tree
  21. 21Boundary Traversal of a Binary Tree
  22. 22Construct a Binary Tree from a String with Bracket Representation
  23. 23Convert a Binary Tree into a Doubly Linked List
  24. 24Convert a Binary Tree into a Sum Tree
  25. 25Find Minimum Swaps Required to Convert a Binary Tree into a BST
  26. 26Check if a Binary Tree is a Sum Tree
  27. 27Check if All Leaf Nodes are at the Same Level in a Binary Tree
  28. 28Lowest Common Ancestor (LCA) in a Binary Tree
  29. 29Solve the Tree Isomorphism Problem
  30. 30Check if a Binary Tree Contains Duplicate Subtrees of Size 2 or More
  31. 31Check if Two Binary Trees are Mirror Images
  32. 32Calculate the Sum of Nodes on the Longest Path from Root to Leaf in a Binary Tree
  33. 33Print All Paths in a Binary Tree with a Given Sum
  34. 34Find the Distance Between Two Nodes in a Binary Tree
  35. 35Find the kth Ancestor of a Node in a Binary Tree
  36. 36Find All Duplicate Subtrees in a Binary Tree
GraphsGraphs46
  1. 1Breadth-First Search in Graphs
  2. 2Depth-First Search in Graphs
  3. 3Number of Provinces in an Undirected Graph
  4. 4Connected Components in a Matrix
  5. 5Rotten Oranges Problem - BFS in Matrix
  6. 6Flood Fill Algorithm - Graph Based
  7. 7Detect Cycle in an Undirected Graph using DFS
  8. 8Detect Cycle in an Undirected Graph using BFS
  9. 9Distance of Nearest Cell Having 1 - Grid BFS
  10. 10Surrounded Regions in Matrix using Graph Traversal
  11. 11Number of Enclaves in Grid
  12. 12Word Ladder - Shortest Transformation using Graph
  13. 13Word Ladder II - All Shortest Transformation Sequences
  14. 14Number of Distinct Islands using DFS
  15. 15Check if a Graph is Bipartite using DFS
  16. 16Topological Sort Using DFS
  17. 17Topological Sort using Kahn's Algorithm
  18. 18Cycle Detection in Directed Graph using BFS
  19. 19Course Schedule - Task Ordering with Prerequisites
  20. 20Course Schedule 2 - Task Ordering Using Topological Sort
  21. 21Find Eventual Safe States in a Directed Graph
  22. 22Alien Dictionary Character Order
  23. 23Shortest Path in Undirected Graph with Unit Distance
  24. 24Shortest Path in DAG using Topological Sort
  25. 25Dijkstra's Algorithm Using Set - Shortest Path in Graph
  26. 26Dijkstra’s Algorithm Using Priority Queue
  27. 27Shortest Distance in a Binary Maze using BFS
  28. 28Path With Minimum Effort in Grid using Graphs
  29. 29Cheapest Flights Within K Stops - Graph Problem
  30. 30Number of Ways to Reach Destination in Shortest Time - Graph Problem
  31. 31Minimum Multiplications to Reach End - Graph BFS
  32. 32Bellman-Ford Algorithm for Shortest Paths
  33. 33Floyd Warshall Algorithm for All-Pairs Shortest Path
  34. 34Find the City With the Fewest Reachable Neighbours
  35. 35Minimum Spanning Tree in Graphs
  36. 36Prim's Algorithm for Minimum Spanning Tree
  37. 37Disjoint Set (Union-Find) with Union by Rank and Path Compression
  38. 38Kruskal's Algorithm - Minimum Spanning Tree
  39. 39Minimum Operations to Make Network Connected
  40. 40Most Stones Removed with Same Row or Column
  41. 41Accounts Merge Problem using Disjoint Set Union
  42. 42Number of Islands II - Online Queries using DSU
  43. 43Making a Large Island Using DSU
  44. 44Bridges in Graph using Tarjan's Algorithm
  45. 45Articulation Points in Graphs
  46. 46Strongly Connected Components using Kosaraju's Algorithm

Find Bridges in Graph Using Tarjan's Algorithm (Time & Low Arrays)

Problem Statement

You are given a network of n servers numbered from 0 to n - 1 connected by undirected server-to-server connections, where each connection is represented as [a, b].

A critical connection is a connection that, if removed, will disconnect the network — making some servers unreachable from others.

Return all such critical connections in any order.

Note: In this problem, servers are treated as nodes of a graph, and connections as undirected edges. The goal is to find all bridges in the graph using Tarjan's Algorithm.

Examples

n Connections Output Description
4 [[0,1],[1,2],[2,0],[1,3]] [[1,3]] Removing edge 1-3 disconnects node 3
5 [[0,1],[1,2],[2,3],[3,4],[4,2]] [[0,1]] Only 0-1 is a bridge; others are in a cycle
2 [[0,1]] [[0,1]] Single connection is always a bridge
3 [[0,1],[1,2],[2,0]] [] Fully connected triangle — no bridges
0 [] [] Empty graph, no edges or bridges

Visualization Player

Solution

Understanding the Problem

We are given a network of servers (nodes) connected by undirected edges. Each connection is a two-way link. Our goal is to find all critical connections, also known as bridges.

A bridge is an edge that, if removed, would increase the number of connected components in the graph. In simple terms, removing that edge disconnects part of the network. This is a classic graph problem, and we can solve it efficiently using Tarjan’s Algorithm, which is based on Depth-First Search (DFS).

Step-by-Step Solution with Example

step 1: Represent the Graph

We first build an adjacency list from the input list of connections. This allows us to efficiently explore all neighbors of a node during DFS.

step 2: Setup Required Structures

We initialize:

  • visited[]: to keep track of visited nodes
  • tin[]: stores the discovery time (insertion time) for each node
  • low[]: the lowest discovery time reachable from that node or its subtree
  • timer: a global counter to assign discovery times
  • result[]: a list to store all bridges found

step 3: Run DFS from Any Node

We begin DFS traversal from an unvisited node. During DFS:

  • We mark the node as visited and assign its tin and low values using the global timer.
  • For each neighbor:
    • If the neighbor is the parent (the node we came from), we skip it.
    • If the neighbor is already visited, it means a back edge exists. We update the current node’s low value using the neighbor’s tin.
    • If the neighbor is unvisited, we recursively apply DFS on it. After returning, we update the current node’s low value based on the neighbor’s low. If low[neighbor] > tin[current], then [current, neighbor] is a bridge.

step 4: Example Walkthrough

Let's consider this input:

n = 5  
connections = [[0,1],[1,2],[2,0],[1,3],[3,4]]

This forms the following graph:

  • A cycle: 0–1–2–0
  • Two edges: 1–3 and 3–4 (which are not part of any cycle)

Running Tarjan's algorithm:

  • DFS visits nodes and assigns timestamps
  • When we backtrack, we find that removing 1–3 or 3–4 would increase the number of components

Output: [[3, 4], [1, 3]]

Edge Cases

  • Disconnected Graph: If the graph has multiple disconnected parts, run DFS for each unvisited node.
  • No Bridges: If the graph is fully connected with cycles (like a complete graph), then no bridge will be found.
  • Single Node: No connections to check, return an empty list.
  • Tree Structure: Every edge in a tree is a bridge since removing any edge disconnects the graph.

Finally

Tarjan’s Algorithm is an elegant and efficient solution to detect bridges in a graph. By tracking the discovery time and the lowest reachable ancestor for each node, we can find edges that, if removed, disconnect the graph. The algorithm runs in O(V + E) time and is highly suitable for large graphs.

Understanding the idea of back edges and how they affect the low value is key to mastering this approach. With step-by-step DFS and careful updates, even beginners can grasp the intuition behind finding critical connections in a network.

Algorithm Steps

  1. Initialize graph as an adjacency list.
  2. Create arrays: tin[], low[] and visited[] of size n.
  3. Set a timer = 0.
  4. For each unvisited node, run DFS:
    1. Mark node as visited, set tin[node] = low[node] = timer++.
    2. For each neighbor:
      1. If it is the parent, continue.
      2. If not visited, recurse and update low[node].
      3. If low[neighbor] > tin[node], then [node, neighbor] is a bridge.
      4. If already visited, update low[node] = min(low[node], tin[neighbor]).

Code

JavaScript
function criticalConnections(n, connections) {
  const graph = Array.from({ length: n }, () => []);
  for (const [u, v] of connections) {
    graph[u].push(v);
    graph[v].push(u);
  }

  const tin = new Array(n).fill(-1);
  const low = new Array(n).fill(-1);
  const visited = new Array(n).fill(false);
  const bridges = [];
  let timer = 0;

  function dfs(node, parent) {
    visited[node] = true;
    tin[node] = low[node] = timer++;

    for (const neighbor of graph[node]) {
      if (neighbor === parent) continue;

      if (!visited[neighbor]) {
        dfs(neighbor, node);
        low[node] = Math.min(low[node], low[neighbor]);

        if (low[neighbor] > tin[node]) {
          bridges.push([node, neighbor]);
        }
      } else {
        low[node] = Math.min(low[node], tin[neighbor]);
      }
    }
  }

  for (let i = 0; i < n; i++) {
    if (!visited[i]) {
      dfs(i, -1);
    }
  }

  return bridges;
}

console.log("Bridges:", criticalConnections(4, [[0,1],[1,2],[2,0],[1,3]]));

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