Bridges in Graph Tarjan's Algorithm in DSA Typescript - Build from Scratch

Bridges in Graph using Tarjan's Algorithm

📖 Scenario: Imagine you are a network engineer checking a computer network. You want to find all the critical connections (called bridges) in the network. A bridge is a connection that, if removed, will split the network into two parts.We will use Tarjan's algorithm to find these bridges in a graph representing the network.

🎯 Goal: Build a TypeScript program that finds all bridges in an undirected graph using Tarjan's algorithm.You will create the graph, set up helper variables, implement the core DFS logic to find bridges, and finally print the list of bridges.

📋 What You'll Learn

Create an adjacency list to represent the graph with exact edges

Create arrays for discovery time and low values with exact names

Implement Tarjan's DFS function with exact variable names

Print the list of bridges exactly as an array of pairs

💡 Why This Matters

🌍 Real World

Network engineers use bridge detection to find critical connections that can cause network failure if removed.

💼 Career

Understanding Tarjan's algorithm and graph traversal is important for roles in network security, infrastructure, and software engineering involving graph data.

1

Create the graph as an adjacency list

Create a variable called graph as an array of arrays of numbers with 5 nodes (0 to 4). Add these edges exactly: 0-1, 1-2, 2-0, 1-3, 3-4.

DSA Typescript

// Your code here

Hint

Use an array of arrays where each index represents a node and contains a list of connected nodes.

2

Create helper arrays and variables

Create arrays called disc and low of length 5 filled with -1. Create a variable time set to 0. Create an empty array bridges to store pairs of numbers.

DSA Typescript

const graph: number[][] = [
  [1, 2],    // edges from node 0
  [0, 2, 3], // edges from node 1
  [0, 1],    // edges from node 2
  [1, 4],    // edges from node 3
  [3]        // edges from node 4
];
// Your code here

Hint

Use new Array(5).fill(-1) to create arrays filled with -1.

3

Implement Tarjan's DFS function to find bridges

Write a function called dfs that takes u and parent as numbers. Inside, set disc[u] and low[u] to time, then increment time. Loop over neighbors v in graph[u]. If disc[v] is -1, call dfs(v, u), update low[u] to Math.min(low[u], low[v]). If low[v] > disc[u], push [u, v] to bridges. Else if v !== parent, update low[u] to Math.min(low[u], disc[v]). Then, run dfs(0, -1).

DSA Typescript

const graph: number[][] = [
  [1, 2],    // edges from node 0
  [0, 2, 3], // edges from node 1
  [0, 1],    // edges from node 2
  [1, 4],    // edges from node 3
  [3]        // edges from node 4
];

const disc: number[] = new Array(5).fill(-1);
const low: number[] = new Array(5).fill(-1);
let time = 0;
const bridges: [number, number][] = [];

// Your code here

Hint

Follow Tarjan's algorithm steps carefully. Use recursion and update disc and low arrays.

4

Print the list of bridges found

Write console.log(bridges) to print the array bridges.

DSA Typescript

const graph: number[][] = [
  [1, 2],    // edges from node 0
  [0, 2, 3], // edges from node 1
  [0, 1],    // edges from node 2
  [1, 4],    // edges from node 3
  [3]        // edges from node 4
];

const disc: number[] = new Array(5).fill(-1);
const low: number[] = new Array(5).fill(-1);
let time = 0;
const bridges: [number, number][] = [];

function dfs(u: number, parent: number): void {
  disc[u] = time;
  low[u] = time;
  time++;

  for (const v of graph[u]) {
    if (disc[v] === -1) {
      dfs(v, u);
      low[u] = Math.min(low[u], low[v]);
      if (low[v] > disc[u]) {
        bridges.push([u, v]);
      }
    } else if (v !== parent) {
      low[u] = Math.min(low[u], disc[v]);
    }
  }
}

dfs(0, -1);

// Your code here

Hint

The bridges in this graph are edges [1,3] and [3,4].

Practice

1. What is the main purpose of Tarjan's algorithm when finding bridges in a graph?

easy

A. To sort nodes in topological order

B. To find the shortest path between two nodes

C. To detect cycles in a graph

D. To find edges that, if removed, increase the number of connected components

Solution

1. Step 1: Understand what a bridge is in a graph

   A bridge is an edge that, when removed, increases the number of connected components in the graph.

2. Step 2: Identify Tarjan's algorithm goal

   Tarjan's algorithm is used to find such edges efficiently by using DFS and tracking discovery and low times.

3. Final Answer:

   To find edges that, if removed, increase the number of connected components -> Option D

4. Quick Check:

   Bridges = edges increasing components when removed [OK]

Hint: Bridges disconnect graph if removed [OK]

Common Mistakes:

• Confusing bridges with cycles
• Thinking it finds shortest paths
• Mixing with topological sorting

2. Which of the following is the correct TypeScript syntax to declare the discovery time array for Tarjan's algorithm?

easy

A. let disc = Array(n).fill('undefined');

B. let disc: number = new Array(n);

C. let disc: number[] = new Array(n).fill(-1);

D. let disc: number[] = new Array(n);

Solution

1. Step 1: Understand array declaration with initial values in TypeScript

   To create an array of numbers with size n and initialize all values to -1, use new Array(n).fill(-1).

2. Step 2: Check type correctness

   TypeScript syntax requires disc to be number[] for an array of numbers.

3. Final Answer:

   let disc: number[] = new Array(n).fill(-1); -> Option C

4. Quick Check:

   Array<number> with fill(-1) = correct initialization [OK]

Hint: Use new Array(n).fill(-1) for initialized arrays [OK]

Common Mistakes:

• Using wrong type annotation
• Not initializing array values
• Filling with string instead of number

3. Consider the following TypeScript code snippet using Tarjan's algorithm to find bridges. What will be the output bridges for the given graph?

const graph = [[1, 2], [0, 2], [0, 1, 3], [2, 4], [3]];
// Edges: 0-1, 0-2, 1-2, 2-3, 3-4
// Find bridges
const bridges = findBridges(graph);
console.log(bridges);

medium

A. [[2, 3], [3, 4]]

B. [[0, 1], [1, 2]]

C. [[0, 2], [2, 3]]

D. []

Solution

1. Step 1: Analyze the graph edges and connectivity

   The graph edges are 0-1, 0-2, 1-2 forming a triangle, and 2-3, 3-4 forming a chain. The triangle edges are not bridges because removing one does not disconnect the graph.

2. Step 2: Identify bridges

   Edges 2-3 and 3-4 are bridges because removing either disconnects node 4 or node 3 from the rest.

3. Final Answer:

   [[2, 3], [3, 4]] -> Option A

4. Quick Check:

   Bridges disconnect graph = [[2,3],[3,4]] [OK]

Hint: Bridges are edges not in any cycle [OK]

Common Mistakes:

• Including edges inside cycles as bridges
• Missing chain edges as bridges
• Confusing node indices

4. The following TypeScript function attempts to find bridges using Tarjan's algorithm but has a bug. What is the bug?

function dfs(u: number, parent: number) {
  visited[u] = true;
  disc[u] = low[u] = time++;
  for (const v of graph[u]) {
    if (!visited[v]) {
      dfs(v, u);
      low[u] = Math.min(low[u], low[v]);
      if (low[v] > disc[u]) {
        bridges.push([u, v]);
      }
    } else if (v !== parent) {
      low[u] = Math.min(low[u], disc[v]);
    }
  }
}

medium

A. The condition 'low[v] > disc[u]' should be 'low[v] >= disc[u]'

B. The 'visited' array is not initialized before DFS

C. The 'time' variable is not incremented correctly

D. The parent node check 'v !== parent' is missing

Solution

1. Step 1: Check initialization of visited array

   The code uses 'visited[u]' but does not show initialization. Without initializing, visited may be undefined causing runtime errors.

2. Step 2: Verify other parts

   The condition 'low[v] > disc[u]' is correct for bridges. The parent check is present. The time increment is correct.

3. Final Answer:

   The 'visited' array is not initialized before DFS -> Option B

4. Quick Check:

   Uninitialized visited array causes errors [OK]

Hint: Always initialize visited before DFS [OK]

Common Mistakes:

• Changing correct bridge condition
• Ignoring visited initialization
• Removing parent check

5. Given a connected undirected graph with nodes 0 to 5 and edges: 0-1, 1-2, 2-0, 1-3, 3-4, 4-5, 5-3, which edges are bridges according to Tarjan's algorithm?

hard

A. [[1, 3]]

B. [[3, 4], [4, 5]]

C. [[0, 1], [1, 2]]

D. []

Solution

1. Step 1: Analyze graph structure

   Edges 0-1, 1-2, 2-0 form a triangle (cycle). Edges 3-4, 4-5, 5-3 form another cycle. Edge 1-3 connects these two cycles.

2. Step 2: Identify bridges

   Only edge 1-3 is a bridge because removing it disconnects the two cycles. Other edges are part of cycles and not bridges.

3. Final Answer:

   [[1, 3]] -> Option A

4. Quick Check:

   Bridge connects two cycles = [[1, 3]] [OK]

Hint: Bridges connect separate cycles or components [OK]

Common Mistakes:

• Marking cycle edges as bridges
• Missing the connecting bridge edge
• Confusing node pairs