Minimum Spanning Tree Kruskal's Algorithm in DSA C - Practice Problems & Challenges

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of Kruskal's MST edges

What edges are included in the Minimum Spanning Tree (MST) after running Kruskal's algorithm on the given graph?

DSA C

Edges with weights: (0-1:4), (0-2:3), (1-2:1), (1-3:2), (2-3:4), (3-4:2), (4-5:6)
Run Kruskal's MST and list edges in MST in order of selection.

A (0-2:3), (1-2:1), (1-3:2), (3-4:2), (4-5:6)

B (1-2:1), (0-1:4), (1-3:2), (3-4:2), (4-5:6)

C (1-3:2), (3-4:2), (0-2:3), (1-2:1), (4-5:6)

D (1-2:1), (1-3:2), (3-4:2), (0-2:3), (4-5:6)

💡 Hint

Sort edges by weight and pick edges that don't form cycles.

Kruskal's algorithm sorts edges by weight and adds edges that connect new vertices without cycles. The edges (1-2:1), (1-3:2), (3-4:2), (0-2:3), and (4-5:6) form the MST.

🧠 Conceptual

intermediate

Why does Kruskal's algorithm use a disjoint set?

What is the main purpose of using a disjoint set (union-find) data structure in Kruskal's algorithm?

A To find the shortest path between two vertices

B To store the graph adjacency list

C To quickly check if adding an edge creates a cycle

D To sort edges by their weights efficiently

💡 Hint

Think about how Kruskal avoids cycles when adding edges.

Disjoint set helps track which vertices are connected. It quickly checks if two vertices belong to the same set, preventing cycles.

🔧 Debug

advanced

Identify the error in this Kruskal's MST code snippet

What error will this code cause when running Kruskal's MST? int parent[5] = {-1, -1, -1, -1, -1}; int find(int x) { if (parent[x] == -1) return x; return find(parent[x]); } void union_set(int x, int y) { int xset = find(x); int yset = find(y); parent[xset] = yset; } // In main, edges are processed and union_set called without checking cycles.

A The code may create cycles because it unions without checking if sets are different

B Stack overflow due to infinite recursion in find function

C Compilation error due to missing return type in union_set

D Array index out of bounds accessing parent array

💡 Hint

Check if union is done only when sets are different.

Union is done without checking if x and y belong to different sets, so cycles can form.

❓ Predict Output

advanced

Output of MST total weight calculation

What is the total weight of the MST after running Kruskal's algorithm on this graph? Edges: (0-1:10), (0-2:6), (0-3:5), (1-3:15), (2-3:4)

A 25

B 19

C 20

D 15

💡 Hint

Add weights of edges selected by Kruskal's MST.

Edges selected: (2-3:4), (0-3:5), (0-1:10). Total weight = 4 + 5 + 10 = 19.

🚀 Application

expert

Kruskal's algorithm on a disconnected graph

If Kruskal's algorithm is run on a graph with two disconnected components, what will be the result?

A It will produce a Minimum Spanning Forest containing MSTs for each component

B It will fail with an error because the graph is not connected

C It will produce a single MST connecting all vertices including disconnected ones

D It will only process the first component and ignore the second

💡 Hint

Think about how MST applies to disconnected graphs.

Kruskal's algorithm produces a Minimum Spanning Forest, which is MSTs for each disconnected component.

Practice

1. What is the main goal of Kruskal's algorithm in graph theory?

easy

A. To sort the vertices of a graph

B. To find the shortest path between two nodes

C. To detect cycles in a graph

D. To find the minimum spanning tree of a graph

Solution

1. Step 1: Understand Kruskal's algorithm purpose

   Kruskal's algorithm is designed to find a subset of edges that connects all vertices with the minimum total edge weight and no cycles.

2. Step 2: Compare with other graph algorithms

   Unlike shortest path or cycle detection, Kruskal's focuses on building a minimum spanning tree.

3. Final Answer:

   To find the minimum spanning tree of a graph -> Option D

4. Quick Check:

   Kruskal's algorithm = Minimum Spanning Tree [OK]

Hint: Kruskal's always builds minimum spanning tree by edges [OK]

Common Mistakes:

• Confusing MST with shortest path
• Thinking it detects cycles only
• Assuming it sorts vertices

2. Which data structure is commonly used in Kruskal's algorithm to detect cycles efficiently?

easy

A. Disjoint Set (Union-Find)

B. Stack

C. Queue

D. Priority Queue

Solution

1. Step 1: Identify cycle detection method in Kruskal's

   Kruskal's algorithm uses a Disjoint Set (Union-Find) data structure to check if adding an edge creates a cycle by checking if two vertices belong to the same set.

2. Step 2: Compare other data structures

   Queue and stack do not help detect cycles efficiently here; priority queue is used for sorting edges but not cycle detection.

3. Final Answer:

   Disjoint Set (Union-Find) -> Option A

4. Quick Check:

   Cycle detection in Kruskal's = Union-Find [OK]

Hint: Use Union-Find to quickly check cycles in Kruskal's [OK]

Common Mistakes:

• Confusing priority queue with cycle detection
• Using stack or queue for cycle detection
• Not knowing Union-Find purpose

3. Consider the following edges of a graph with weights:
Edges = [(A-B, 1), (B-C, 4), (A-C, 3), (C-D, 2)]
What is the total weight of the minimum spanning tree found by Kruskal's algorithm?

medium

A. 7

B. 6

C. 8

D. 10

Solution

1. Step 1: Sort edges by weight

   Sorted edges: (A-B,1), (C-D,2), (A-C,3), (B-C,4)

2. Step 2: Pick edges without cycles

   Pick (A-B,1), then (C-D,2), then (A-C,3). Adding (B-C,4) creates a cycle, so skip it.

3. Step 3: Calculate total weight

   Total weight = 1 + 2 + 3 = 6

4. Final Answer:

   6 -> Option B

5. Quick Check:

   Sum of MST edges = 6 [OK]

Hint: Sort edges, pick smallest without cycle until all connected [OK]

Common Mistakes:

• Adding all edges without checking cycles
• Choosing edges in wrong order
• Miscounting total weight

4. The following C code snippet is part of Kruskal's algorithm. What is the error?

int findParent(int parent[], int i) {
    if (parent[i] == i)
        return i;
    return findParent(parent, parent[i]);
}

void unionSet(int parent[], int x, int y) {
    int xset = findParent(parent, x);
    int yset = findParent(parent, y);
    parent[xset] = yset;
}

medium

A. unionSet should swap xset and yset

B. unionSet should not change parent array

C. Missing path compression in findParent

D. findParent should return void

Solution

1. Step 1: Analyze findParent function

   findParent recursively finds the root parent but does not update parent[i] to root, missing path compression optimization.

2. Step 2: Understand unionSet function

   unionSet correctly merges sets by updating parent of one root to another; no swap needed.

3. Final Answer:

   Missing path compression in findParent -> Option C

4. Quick Check:

   Path compression missing = findParent optimization [OK]

Hint: Add path compression to speed up findParent [OK]

Common Mistakes:

• Thinking unionSet must swap parents
• Believing findParent should return void
• Ignoring path compression importance

5. Given a graph with vertices {1, 2, 3, 4} and edges with weights:
(1-2, 5), (1-3, 3), (2-3, 4), (2-4, 2), (3-4, 7)
Using Kruskal's algorithm, which edges will be included in the minimum spanning tree?

hard

A. (2-4), (1-3), (2-3)

B. (1-3), (2-4), (1-2)

C. (2-4), (1-3), (1-2)

D. (1-2), (2-3), (3-4)

Solution

1. Step 1: Sort edges by weight

   Sorted edges: (2-4,2), (1-3,3), (2-3,4), (1-2,5), (3-4,7)

2. Step 2: Pick edges without cycles

   Pick (2-4,2), then (1-3,3). Next (2-3,4) connects 2 and 3 which are in different sets, so pick it. After these three edges, all vertices are connected and MST is complete.

3. Step 3: Verify options

   (2-4), (1-3), (2-3) matches edges (2-4), (1-3), (2-3). (2-4), (1-3), (1-2) includes (1-2) which is heavier than (2-3), so incorrect.

4. Final Answer:

   (2-4), (1-3), (2-3) -> Option A

5. Quick Check:

   Pick smallest edges connecting all vertices without cycle [OK]

Hint: Sort edges, pick smallest connecting new vertices until all connected [OK]

Common Mistakes:

• Choosing heavier edges over lighter ones
• Adding edges that create cycles
• Not stopping after n-1 edges