Bridges in Graph Tarjan's Algorithm in DSA Typescript - Cheat Sheet & Quick Revision

Recall & Review

beginner

What is a bridge in a graph?

A bridge is an edge in a graph that, if removed, increases the number of connected components. It means the edge is critical for keeping the graph connected.

intermediate

What is the main idea behind Tarjan's algorithm for finding bridges?

Tarjan's algorithm uses a depth-first search (DFS) to assign discovery times and low values to each vertex. An edge is a bridge if the low value of the connected vertex is greater than the discovery time of the current vertex.

intermediate

In Tarjan's algorithm, what does the 'low' value represent?

The 'low' value of a vertex is the earliest discovery time reachable from that vertex or its descendants by following zero or more tree edges and then possibly one back edge.

advanced

Why do we compare low[v] > disc[u] to find a bridge between u and v?

If low[v] > disc[u], it means vertex v and its descendants cannot reach any ancestor of u without using the edge (u, v). So, removing (u, v) disconnects the graph, making it a bridge.

beginner

What is the time complexity of Tarjan's algorithm for finding bridges?

The time complexity is O(V + E), where V is the number of vertices and E is the number of edges, because it performs a single DFS traversal.

What does a bridge in a graph do?

A Connects two disconnected graphs

B Decreases the number of connected components if removed

C Has no effect on connectivity

D Increases the number of connected components if removed

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

In Tarjan's algorithm, what is the purpose of the discovery time array?

A To count the number of edges

B To store the degree of each vertex

C To store the order in which vertices are visited in DFS

D To mark visited vertices

Discovery time records the order in which vertices are first visited during DFS.

What does it mean if low[v] <= disc[u] in Tarjan's algorithm?

A Edge (u, v) is not a bridge

B Edge (u, v) is a bridge

C Vertex v is not reachable

D Vertex u is a leaf

If low[v] <= disc[u], it means v or its descendants can reach u or its ancestors, so (u, v) is not a bridge.

Which traversal method does Tarjan's algorithm use?

A Depth-first search

B Dijkstra's algorithm

C Breadth-first search

D Prim's algorithm

Tarjan's algorithm uses depth-first search to explore the graph.

What is the time complexity of Tarjan's algorithm for bridges?

A O(E^2)

B O(V + E)

C O(V^2)

D O(log V)

Tarjan's algorithm runs in linear time relative to vertices and edges, O(V + E).

Explain how Tarjan's algorithm finds bridges in a graph using discovery and low values.

Think about how DFS helps track back edges and earliest reachable ancestors.

Describe why an edge is considered a bridge in terms of graph connectivity and Tarjan's algorithm.

Focus on the meaning of low and discovery times in connectivity.

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