Bridges in Graph Tarjan's Algorithm in DSA Java - Interactive Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to initialize the discovery time for the current node in Tarjan's algorithm.

DSA Java

disc[u] = [1];

Drag options to blanks, or click blank then click option'

Atime

Bu

C-1

D0

Attempts:

3 left

💡 Hint

Common Mistakes

Setting discovery time to node index instead of time counter.

Initializing discovery time to zero or -1 incorrectly.

In Tarjan's algorithm, the discovery time of a node is set to the current time counter.

2fill in blank

medium

Complete the code to update the low time of the current node after visiting a neighbor.

DSA Java

low[u] = Math.min(low[u], [1]);

Drag options to blanks, or click blank then click option'

Adisc[v]

Bu

Ctime

Dlow[v]

Attempts:

3 left

💡 Hint

Common Mistakes

Using discovery time of neighbor instead of low time.

Using the current node's discovery time incorrectly.

The low time of the current node is updated with the minimum low time of its neighbors.

3fill in blank

hard

Fix the error in the condition that checks if an edge is a bridge.

DSA Java

if (low[[1]] > disc[u]) {

Drag options to blanks, or click blank then click option'

Atime

Bparent[u]

Cv

Du

Attempts:

3 left

💡 Hint

Common Mistakes

Using u instead of v in the low time comparison.

Comparing with parent or time variables incorrectly.

The condition checks if the low time of neighbor v is greater than the discovery time of u, indicating a bridge.

4fill in blank

hard

Fill both blanks to correctly skip the parent node during DFS traversal.

DSA Java

if (v == [1]) continue;
parent[[2]] = u;

Drag options to blanks, or click blank then click option'

A-1

Bparent[u]

Cu

Dv

Attempts:

3 left

💡 Hint

Common Mistakes

Skipping the wrong node instead of parent[u].

Setting parent of u instead of v.

We skip the parent of u to avoid revisiting it, and set the parent of v as u during DFS.

5fill in blank

hard

Fill all three blanks to correctly implement the DFS call and update low time after recursion.

DSA Java

dfs([1], [2], disc, low, parent, graph, time);
low[u] = Math.min(low[u], [3]);

Drag options to blanks, or click blank then click option'

Av

Bu

Clow[v]

Dparent

Attempts:

3 left

💡 Hint

Common Mistakes

Swapping u and v in the DFS call.

Updating low[u] with disc[v] instead of low[v].

We call DFS on neighbor v with current node u as parent, then update low[u] with low[v].

Practice

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

easy

A. To identify edges that, if removed, increase the number of connected components

B. To find the shortest path between two nodes

C. To detect cycles in a graph

D. To sort nodes in topological order

Solution

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

   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 identify edges that, if removed, increase the number of connected components -> Option A

4. Quick Check:

   Bridges = edges increasing components [OK]

Hint: Bridges split graph into more parts when removed [OK]

Common Mistakes:

• Confusing bridges with cycles
• Thinking it finds shortest paths
• Mixing bridges with articulation points

2. Which of the following Java code snippets correctly initializes the arrays used for discovery and low times in Tarjan's algorithm for a graph with n nodes?

easy

A. int[] disc = new int[n]; int[] low = new int[n]; Arrays.fill(disc, 0); Arrays.fill(low, 0);

B. int[] disc = new int[n]; int[] low = new int[n];

C. int disc[] = new int[n]; int low[] = new int[n]; disc = null; low = null;

D. int[] disc = new int[n]; int[] low = new int[n]; Arrays.fill(disc, -1); Arrays.fill(low, -1);

Solution

1. Step 1: Check initialization of discovery and low arrays

   Both arrays must be initialized with -1 to indicate unvisited nodes.

2. Step 2: Verify correct Java syntax and usage

   int[] disc = new int[n]; int[] low = new int[n]; Arrays.fill(disc, -1); Arrays.fill(low, -1); correctly creates arrays and fills them with -1 using Arrays.fill.

3. Final Answer:

   int[] disc = new int[n]; int[] low = new int[n]; Arrays.fill(disc, -1); Arrays.fill(low, -1); -> Option D

4. Quick Check:

   Arrays initialized with -1 = int[] disc = new int[n]; int[] low = new int[n]; Arrays.fill(disc, -1); Arrays.fill(low, -1); [OK]

Hint: Initialize discovery and low arrays with -1 for unvisited [OK]

Common Mistakes:

• Not filling arrays with -1, causing wrong visited checks
• Setting arrays to null after creation
• Filling arrays with 0 instead of -1

3. Consider the following Java code snippet implementing part of Tarjan's algorithm for bridges detection. What will be the output list of bridges for the given graph edges?

int n = 5;
List<List<Integer>> graph = new ArrayList<>();
for (int i = 0; i < n; i++) graph.add(new ArrayList<>());
graph.get(0).add(1); graph.get(1).add(0);
graph.get(1).add(2); graph.get(2).add(1);
graph.get(2).add(0); graph.get(0).add(2);
graph.get(1).add(3); graph.get(3).add(1);
graph.get(3).add(4); graph.get(4).add(3);

// After running Tarjan's bridge detection, which edges are bridges?

medium

A. [[0, 1], [1, 2]]

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

C. [[1, 3], [3, 4]]

D. []

Solution

1. Step 1: Analyze the graph structure

   The graph has edges forming a triangle between nodes 0,1,2 and a chain 1-3-4.

2. Step 2: Identify bridges

   Edges in the triangle are not bridges because removing them doesn't disconnect the graph. Edges (1,3) and (3,4) are bridges because removing them increases components.

3. Final Answer:

   [[1, 3], [3, 4]] -> Option C

4. Quick Check:

   Bridges separate graph parts = [[1, 3], [3, 4]] [OK]

Hint: Edges in cycles are not bridges; chains are [OK]

Common Mistakes:

• Marking edges in cycles as bridges
• Ignoring edges that disconnect graph
• Confusing node order in edges

4. The following Java code snippet is intended to find bridges using Tarjan's algorithm but contains a bug. What is the bug?

void dfs(int u, int parent) {
    disc[u] = time;
    low[u] = time;
    for (int v : graph.get(u)) {
        if (v == parent) continue;
        if (disc[v] == -1) {
            dfs(v, u);
            low[u] = Math.min(low[u], low[v]);
            if (low[v] > disc[u]) {
                bridges.add(new int[]{u, v});
            }
        } else {
            low[u] = Math.min(low[u], disc[v]);
        }
    }
}

medium

A. The time variable is not incremented correctly

B. The condition to add a bridge should be low[v] >= disc[u]

C. The else block should update low[u] with low[v], not disc[v]

D. The parent check should be if (v != parent) instead of if (v == parent)

Solution

1. Step 1: Check time increment usage

   The code assigns the current time to disc[u] and low[u] but fails to increment the time variable. This causes all nodes to receive the same discovery and low values, breaking the algorithm.

2. Step 2: Verify other potential issues

   The bridge condition (low[v] > disc[u]) is correct. The else block properly updates low[u] with disc[v] for back edges. The parent skip is correctly implemented.

3. Final Answer:

   The time variable is not incremented correctly -> Option A

4. Quick Check:

   Time increment missing [OK]

Hint: Check time increments carefully in DFS [OK]

Common Mistakes:

• Changing bridge condition incorrectly
• Updating low[u] with wrong values
• Misusing parent check condition

5. Given a connected undirected graph with n nodes and m edges, you want to find all bridges using Tarjan's algorithm. Which of the following modifications will correctly handle disconnected graphs as well?

hard

A. Only run DFS from node 0 since graph is connected

B. Run DFS from every node that is unvisited to cover all components

C. Reset discovery and low arrays to zero before each DFS call

D. Increase the time variable by 2 after each DFS call

Solution

1. Step 1: Understand the problem with disconnected graphs

   Tarjan's algorithm assumes DFS covers all nodes. For disconnected graphs, DFS from one node misses others.

2. Step 2: Correct approach for disconnected graphs

   Run DFS from every unvisited node to ensure all components are covered and bridges found.

3. Final Answer:

   Run DFS from every node that is unvisited to cover all components -> Option B

4. Quick Check:

   DFS from all unvisited nodes = Run DFS from every node that is unvisited to cover all components [OK]

Hint: Run DFS on all unvisited nodes for disconnected graphs [OK]

Common Mistakes:

• Assuming graph is always connected
• Resetting arrays incorrectly
• Changing time increments unnecessarily