Minimum Spanning Tree Kruskal's Algorithm in DSA Java - Interactive Practice

Practice - 5 Tasks

1fill in blank

easy

Complete the code to declare the Edge class with weight attribute.

DSA Java

class Edge implements Comparable<Edge> {
    int src, dest, weight;
    Edge(int s, int d, int w) {
        src = s;
        dest = d;
        [1] = w;
    }

    public int compareTo(Edge other) {
        return this.weight - other.weight;
    }
}

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

Adest

Bw

Cweight

Dsrc

Attempts:

3 left

💡 Hint

Common Mistakes

Assigning to the wrong variable like 'dest = w;'

Using the parameter name instead of the instance variable

The constructor assigns the parameter w to the instance variable weight.

2fill in blank

medium

Complete the code to initialize the parent array for union-find.

DSA Java

int[] parent = new int[V];
for (int i = 0; i < V; i++) {
    parent[i] = [1];
}

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

Ai

B-1

C0

DV

Attempts:

3 left

💡 Hint

Common Mistakes

Setting all parents to i or 0 which breaks union-find logic

Each vertex is initially its own parent, so parent[i] = -1.

3fill in blank

hard

Fix the error in the find method to correctly find the root parent.

DSA Java

int find(int[] parent, int i) {
    if (parent[i] == [1]) {
        return i;
    }
    return find(parent, parent[i]);
}

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

A-1

Bi

Cparent[i]

D0

Attempts:

3 left

💡 Hint

Common Mistakes

Checking parent[i] == i which is incorrect for this union-find implementation

In union-find, root parents have parent[i] == -1.

4fill in blank

hard

Fill both blanks to correctly perform union operation.

DSA Java

void union(int[] parent, int x, int y) {
    int xset = find(parent, x);
    int yset = find(parent, y);
    if (xset != yset) {
        parent[[1]] = [2];
    }
}

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

Axset

Byset

Cx

Dy

Attempts:

3 left

💡 Hint

Common Mistakes

Using x or y instead of root parents xset or yset

Assigning parent[yset] = xset instead of parent[xset] = yset (both valid but must be consistent)

Union sets by making one root parent point to the other root parent.

5fill in blank

hard

Fill all three blanks to complete the Kruskal's algorithm loop for MST.

DSA Java

int e = 0; // count of edges in MST
int i = 0; // index for sorted edges
while (e < V - 1 && i < edges.length) {
    Edge nextEdge = edges[[1]];
    int x = find(parent, nextEdge.src);
    int y = find(parent, nextEdge.dest);
    if (x != y) {
        union(parent, x, y);
        mstEdges[[2]] = nextEdge;
        e[3];
    }
    i++;
}

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

Ai

Be

C++

D--

Attempts:

3 left

💡 Hint

Common Mistakes

Using wrong indices for arrays

Decrementing e instead of incrementing

Use i to index edges, e to index MST edges, and increment e after adding an edge.

Practice

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

easy

A. To detect cycles in a graph

B. To find the shortest path between two nodes

C. To find the minimum spanning tree of a graph

D. To sort the vertices 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 problems

   Shortest path and cycle detection are different problems solved by other algorithms like Dijkstra or DFS.

3. Final Answer:

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

4. Quick Check:

   Minimum Spanning Tree = Find minimum edges connecting all nodes [OK]

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

Common Mistakes:

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

2. Which Java data structure is commonly used to efficiently check and union sets in Kruskal's algorithm?

easy

A. Disjoint Set (Union-Find)

B. PriorityQueue

C. ArrayList

D. HashMap

Solution

1. Step 1: Identify the need for set operations

   Kruskal's algorithm requires checking if two vertices belong to the same set and merging sets without cycles.

2. Step 2: Match data structure to operations

   Disjoint Set (Union-Find) supports fast union and find operations, ideal for this purpose.

3. Final Answer:

   Disjoint Set (Union-Find) -> Option A

4. Quick Check:

   Union-Find = fast set checks and merges [OK]

Hint: Use Union-Find to detect cycles fast in Kruskal [OK]

Common Mistakes:

• Using ArrayList for set operations
• Confusing PriorityQueue with Union-Find
• Using HashMap which is not efficient here

3. What is the output MST edges after running Kruskal's algorithm on this graph?

Vertices: 1, 2, 3, 4
Edges with weights:
1-2: 1
1-3: 3
2-3: 2
3-4: 4
2-4: 5

medium

A. [1-2, 2-3, 3-4]

B. [1-3, 2-3, 3-4]

C. [1-2, 1-3, 2-4]

D. [1-2, 2-4, 3-4]

Solution

1. Step 1: Sort edges by weight

   Edges sorted: 1-2(1), 2-3(2), 1-3(3), 3-4(4), 2-4(5)

2. Step 2: Pick edges without cycles

   Pick 1-2, then 2-3 (no cycle), then 1-3 creates cycle so skip, pick 3-4 (no cycle), skip 2-4 (cycle)

3. Final Answer:

   [1-2, 2-3, 3-4] -> Option A

4. Quick Check:

   Edges chosen by increasing weight without cycles [OK]

Hint: Sort edges by weight, pick smallest without cycle [OK]

Common Mistakes:

• Picking edges that form cycles
• Not sorting edges by weight
• Choosing heavier edges first

4. Identify the error in this Java snippet for Kruskal's algorithm:

for (Edge e : edges) {
  if (findSet(e.u) != findSet(e.v)) {
    union(e.u, e.v);
    mst.add(e);
  }
}
Collections.sort(edges);

medium

A. Union should be called before findSet

B. Edges should be sorted before the loop

C. mst.add(e) should be outside the if condition

D. The loop should iterate over mst, not edges

Solution

1. Step 1: Check order of sorting and iteration

   Kruskal's requires edges sorted by weight before processing to pick smallest edges first.

2. Step 2: Identify mistake in code order

   Sorting is done after the loop, so edges are not processed in correct order.

3. Final Answer:

   Edges should be sorted before the loop -> Option B

4. Quick Check:

   Sort edges first, then pick edges [OK]

Hint: Sort edges before loop to pick smallest first [OK]

Common Mistakes:

• Sorting edges after processing
• Calling union before checking sets
• Adding edges outside condition

5. Given a graph with edges:

Edges: (A-B, 4), (A-C, 3), (B-C, 2), (B-D, 5), (C-D, 7), (C-E, 8), (D-E, 6)

Using Kruskal's algorithm, what is the total weight of the minimum spanning tree?

hard

A. 18

B. 14

C. 20

D. 16

Solution

1. Step 1: Sort edges by weight

   Sorted edges: B-C(2), A-C(3), A-B(4), B-D(5), D-E(6), C-D(7), C-E(8)

2. Step 2: Pick edges without cycles until all vertices connected

   Pick B-C(2), A-C(3), A-B(4) forms cycle so skip, pick B-D(5), pick D-E(6). Total edges chosen: B-C, A-C, B-D, D-E.

3. Step 3: Calculate total weight

   2 + 3 + 5 + 6 = 16. But A-B(4) skipped due to cycle, so MST edges are B-C, A-C, B-D, D-E.

4. Step 4: Re-check cycle with A-B(4)

   Actually, A-B(4) creates cycle with A-C and B-C, so skip it. MST weight is 2+3+5+6=16.

5. Final Answer:

   16 -> Option D

6. Quick Check:

   Sum of chosen edges weights = 16 [OK]

Hint: Sort edges, pick smallest without cycle, sum weights [OK]

Common Mistakes:

• Including edges that form cycles
• Misordering edges by weight
• Adding all edges without checking cycles