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

Practice - 5 Tasks

1fill in blank

easy

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

DSA Javascript

const parent = Array(n).fill().map((_, i) => [1]);

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

A0

B-1

Ci

Dnull

Attempts:

3 left

💡 Hint

Common Mistakes

Setting all parents to 0 or -1 causes incorrect union-find behavior.

Each node is initially its own parent, so parent[i] = i.

2fill in blank

medium

Complete the code to find the root parent of a node with path compression.

DSA Javascript

function find(x) {
  if (parent[x] !== x) {
    parent[x] = [1](parent[x]);
  }
  return parent[x];
}

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

Aunion

Bfind

Csort

Dpush

Attempts:

3 left

💡 Hint

Common Mistakes

Using union or other functions instead of find causes errors.

The find function calls itself recursively to find the root parent.

3fill in blank

hard

Fix the error in the union function to correctly merge two sets.

DSA Javascript

function union(a, b) {
  a = find(a);
  b = find(b);
  if (a !== b) {
    parent[[1]] = [2];
    return true;
  }
  return false;
}

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

Aa

Bb

Cparent[a]

Dparent[b]

Attempts:

3 left

💡 Hint

Common Mistakes

Assigning parent[a] to parent[b] instead of parent[a] to b.

Mixing up a and b in assignment.

We set parent of root a to root b to merge sets.

4fill in blank

hard

Fill both blanks to sort edges by weight and iterate over them.

DSA Javascript

edges.[1]((a, b) => a.[2] - b.[2]);
for (const edge of edges) {
  // process edge
}

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

Asort

Bweight

Creverse

Dlength

Attempts:

3 left

💡 Hint

Common Mistakes

Using reverse instead of sort.

Comparing by length or other properties.

We sort edges by their weight property using sort and weight.

5fill in blank

hard

Fill all three blanks to build the MST using union-find and accumulate total weight.

DSA Javascript

let mstWeight = 0;
for (const [1] of edges) {
  if (union([2], [3])) {
    mstWeight += edge.weight;
  }
}

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

Aedge

Bedge.u

Cedge.v

Dedges

Attempts:

3 left

💡 Hint

Common Mistakes

Using edges instead of edge in the loop.

Mixing up edge.u and edge.v.

We iterate over each edge, union its nodes u and v, and add weight if union succeeds.

Practice

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

easy

A. To find the minimum spanning tree of a graph

B. To find the shortest path between two nodes

C. To detect cycles in 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, not Kruskal's.

3. Final Answer:

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

4. Quick Check:

   Minimum Spanning Tree = Kruskal's Algorithm goal [OK]

Hint: Kruskal connects all nodes with least total edge weight [OK]

Common Mistakes:

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

2. Which of the following is the correct way to sort edges by weight in JavaScript for Kruskal's Algorithm?

easy

A. edges.sort((a, b) => a.weight * b.weight);

B. edges.sort((a, b) => b.weight - a.weight);

C. edges.sort((a, b) => a.weight + b.weight);

D. edges.sort((a, b) => a.weight - b.weight);

Solution

1. Step 1: Understand sorting for MST

   Kruskal's Algorithm requires edges sorted in ascending order by weight to pick smallest edges first.

2. Step 2: Check JavaScript sort syntax

   Using edges.sort((a, b) => a.weight - b.weight); sorts edges from smallest to largest weight correctly.

3. Final Answer:

   edges.sort((a, b) => a.weight - b.weight); -> Option D

4. Quick Check:

   Sort ascending by weight = edges.sort((a, b) => a.weight - b.weight); [OK]

Hint: Sort edges ascending by weight using (a,b) => a.weight - b.weight [OK]

Common Mistakes:

• Sorting descending instead of ascending
• Using addition or multiplication in sort callback
• Not returning a number in sort function

3. Consider the following edges of a graph:
const edges = [ {from: 'A', to: 'B', weight: 1}, {from: 'B', to: 'C', weight: 4}, {from: 'A', to: 'C', weight: 3}, {from: 'C', to: 'D', weight: 2} ];
After applying Kruskal's Algorithm, which edges will be included in the MST?

medium

A. [{A-B,1}, {A-C,3}, {B-C,4}]

B. [{A-B,1}, {C-D,2}, {A-C,3}]

C. [{A-B,1}, {C-D,2}, {B-C,4}]

D. [{B-C,4}, {A-C,3}, {C-D,2}]

Solution

1. Step 1: Sort edges by weight ascending

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

2. Step 2: Pick edges without forming cycles

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

3. Final Answer:

   [{A-B,1}, {C-D,2}, {A-C,3}] -> Option B

4. Quick Check:

   Edges with smallest weights and no cycles = [{A-B,1}, {C-D,2}, {A-C,3}] [OK]

Hint: Pick edges ascending weight, skip if cycle forms [OK]

Common Mistakes:

• Including edges that form cycles
• Not sorting edges before selection
• Choosing heavier edges unnecessarily

4. The following code snippet tries to implement Kruskal's Algorithm but has a bug:

const edges = [
  {from: 'A', to: 'B', weight: 1},
  {from: 'B', to: 'C', weight: 2},
  {from: 'A', to: 'C', weight: 3}
];

edges.sort((a, b) => a.weight - b.weight);

const mst = [];
const visited = new Set();

for (const edge of edges) {
  if (!visited.has(edge.from) || !visited.has(edge.to)) {
    mst.push(edge);
    visited.add(edge.from);
    visited.add(edge.to);
  }
}
console.log(mst);

What is the main issue with this code?

medium

A. It does not correctly check for cycles, so MST may be incorrect

B. The edges are not sorted properly

C. The visited set is not initialized

D. It uses an array instead of a linked list for MST

Solution

1. Step 1: Analyze cycle detection logic

   The code uses a visited set to check if nodes are visited, but this does not detect cycles correctly in Kruskal's Algorithm.

2. Step 2: Understand correct cycle detection

   Kruskal's requires a Union-Find or Disjoint Set data structure to detect cycles properly, not just visited nodes.

3. Final Answer:

   It does not correctly check for cycles, so MST may be incorrect -> Option A

4. Quick Check:

   Cycle detection needs Union-Find, not visited set [OK]

Hint: Use Union-Find to detect cycles, not visited nodes [OK]

Common Mistakes:

• Using visited set for cycle detection
• Ignoring cycle checks
• Assuming sorting fixes all issues

5. You have a graph with vertices ['A', 'B', 'C', 'D', 'E'] and edges:

const edges = [
  {from: 'A', to: 'B', weight: 1},
  {from: 'B', to: 'C', weight: 3},
  {from: 'A', to: 'C', weight: 2},
  {from: 'C', to: 'D', weight: 4},
  {from: 'D', to: 'E', weight: 5},
  {from: 'B', to: 'E', weight: 6}
];

Using Kruskal's Algorithm with a Union-Find structure, what is the total weight of the minimum spanning tree?

hard

A. 10

B. 15

C. 12

D. 16

Solution

1. Step 1: Sort edges by weight ascending

   Sorted edges: (A-B,1), (A-C,2), (B-C,3), (C-D,4), (D-E,5), (B-E,6)

2. Step 2: Pick edges without cycles using Union-Find

   Pick A-B(1), A-C(2), B-C(3) forms cycle, skip B-C. Pick C-D(4), D-E(5). Total weight = 1+2+4+5 = 12.

3. Final Answer:

   12 -> Option C

4. Quick Check:

   Sum of MST edges = 12 [OK]

Hint: Sum smallest edges without cycles using Union-Find [OK]

Common Mistakes:

• Including edges that form cycles
• Adding all edges without checking
• Misordering edges before selection