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

Practice - 5 Tasks

1fill in blank

easy

Complete the code to initialize the parent array for Union-Find.

DSA Typescript

const parent: number[] = new Array(n).fill(0).map((_, i) => [1]);

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

Ai

B0

Cn

D-1

Attempts:

3 left

💡 Hint

Common Mistakes

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

Using n or -1 as parent values leads to errors.

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

2fill in blank

medium

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

DSA Typescript

function findParent(parent: number[], node: number): number {
  if (parent[node] !== node) {
    parent[node] = [1];
  }
  return parent[node];
}

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

Anode + 1

Bnode

Cparent[node]

DfindParent(parent, parent[node])

Attempts:

3 left

💡 Hint

Common Mistakes

Assigning parent[node] to node itself breaks path compression.

Using parent[node] without recursion does not find root.

Path compression sets parent[node] to the root parent found recursively.

3fill in blank

hard

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

DSA Typescript

function union(parent: number[], a: number, b: number): void {
  const rootA = findParent(parent, a);
  const rootB = findParent(parent, b);
  if (rootA !== rootB) {
    parent[[1]] = rootB;
  }
}

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

ArootA

Bb

Ca

DrootB

Attempts:

3 left

💡 Hint

Common Mistakes

Assigning parent of a or b instead of root parents causes incorrect unions.

Assigning parent[rootB] to rootA reverses union direction.

We set parent of rootA to rootB to union the sets correctly.

4fill in blank

hard

Fill both blanks to correctly check if two nodes belong to different sets before union.

DSA Typescript

if (findParent(parent, [1]) !== findParent(parent, [2])) {
  union(parent, u, v);
}

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

Au

Bv

Cparent[u]

Dparent[v]

Attempts:

3 left

💡 Hint

Common Mistakes

Using parent[u] or parent[v] instead of nodes causes wrong comparisons.

Comparing u and v directly without findParent misses set roots.

We compare root parents of u and v to decide if union is needed.

5fill in blank

hard

Fill all three blanks to sort edges by weight and build MST using Kruskal's algorithm.

DSA Typescript

edges.sort((a, b) => a[1]b[2]);
const mst: Edge[] = [];
for (const {u, v, weight} of edges) {
  if (findParent(parent, u) !== findParent(parent, v)) {
    union(parent, u, v);
    mst[3]({u, v, weight});
  }
}

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

A.weight -

B.weight >

C.weight <

Dpush

Attempts:

3 left

💡 Hint

Common Mistakes

Using comparison operators instead of subtraction in sort causes incorrect sorting.

Using incorrect method instead of push to add edges.

We sort edges by ascending weight using a subtraction comparator, check sets, and push edges to MST.

Practice

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

easy

A. To find the minimum spanning tree of a connected weighted 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 the purpose of Kruskal's Algorithm

   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 algorithms, Kruskal's focuses on building a minimum spanning tree.

3. Final Answer:

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

4. Quick Check:

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

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

Common Mistakes:

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

2. Which of the following TypeScript code snippets correctly initializes the edges array for Kruskal's Algorithm?

easy

A. const edges: number[][] = [0, 1, 10, 1, 2, 5, 0, 2, 15];

B. const edges = {0:1:10, 1:2:5, 0:2:15};

C. const edges: [number, number, number][] = [[0, 1, 10], [1, 2, 5], [0, 2, 15]];

D. const edges: Array<number> = [[0, 1, 10], [1, 2, 5], [0, 2, 15]];

Solution

1. Step 1: Check the correct TypeScript tuple array syntax

   Edges should be an array of tuples, each tuple having three numbers: [node1, node2, weight]. const edges: [number, number, number][] = [[0, 1, 10], [1, 2, 5], [0, 2, 15]]; uses this syntax correctly.

2. Step 2: Identify incorrect syntax in other options

   const edges = {0:1:10, 1:2:5, 0:2:15}; uses invalid object syntax. const edges: number[][] = [0, 1, 10, 1, 2, 5, 0, 2, 15]; is a flat array, not an array of tuples. const edges: Array<number> = [[0, 1, 10], [1, 2, 5], [0, 2, 15]]; uses wrong type annotation (Array<number> instead of tuple array).

3. Final Answer:

   const edges: [number, number, number][] = [[0, 1, 10], [1, 2, 5], [0, 2, 15]]; -> Option C

4. Quick Check:

   Tuple array syntax = const edges: [number, number, number][] = [[0, 1, 10], [1, 2, 5], [0, 2, 15]]; [OK]

Hint: Use tuple arrays for edges: [node1, node2, weight][] [OK]

Common Mistakes:

• Using flat arrays instead of tuples
• Wrong object syntax for edges
• Incorrect type annotations

3. What will be the output edges of the minimum spanning tree after running Kruskal's Algorithm on this graph?

const edges: [number, number, number][] = [[0, 1, 4], [0, 2, 3], [1, 2, 1], [1, 3, 2], [2, 3, 4], [3, 4, 2], [4, 5, 6]];

medium

A. [[0, 1, 4], [1, 3, 2], [3, 4, 2], [4, 5, 6]]

B. [[1, 2, 1], [1, 3, 2], [3, 4, 2], [0, 2, 3], [4, 5, 6]]

C. [[1, 2, 1], [0, 2, 3], [1, 3, 2], [3, 4, 2]]

D. [[0, 1, 4], [0, 2, 3], [1, 2, 1], [4, 5, 6]]

Solution

1. Step 1: Sort edges by weight

   Sorted edges: [1,2,1], [1,3,2], [3,4,2], [0,2,3], [0,1,4], [2,3,4], [4,5,6]

2. Step 2: Pick edges without forming cycles until all nodes connected

   Pick [1,2,1], [1,3,2], [3,4,2], [0,2,3], and [4,5,6]. These connect all nodes with minimum total weight without cycles.

3. Final Answer:

   [[1, 2, 1], [1, 3, 2], [3, 4, 2], [0, 2, 3], [4, 5, 6]] -> Option B

4. Quick Check:

   Edges sorted + no cycles = [[1, 2, 1], [1, 3, 2], [3, 4, 2], [0, 2, 3], [4, 5, 6]] [OK]

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

Common Mistakes:

• Choosing heavier edges first
• Including edges that form cycles
• Missing edges to connect all nodes

4. Identify the error in this TypeScript snippet implementing Kruskal's Algorithm:

const edges: [number, number, number][] = [[0,1,1],[1,2,2],[0,2,3]];
const parent = [0,1,2];
function find(x: number): number {
  if(parent[x] === x) return x;
  return find(parent[x]);
}
function union(a: number, b: number) {
  const rootA = find(a);
  const rootB = find(b);
  if(rootA !== rootB) parent[rootB] = rootA;
}
// Missing sorting of edges before processing

medium

A. The union function incorrectly assigns parent

B. The find function does not implement path compression

C. The parent array should be initialized with -1

D. The edges array is not sorted by weight before processing

Solution

1. Step 1: Check Kruskal's Algorithm requirements

   Kruskal's requires edges to be sorted by weight ascending before union-find operations.

2. Step 2: Identify missing sorting step

   The code comment notes missing sorting; without sorting, MST may not be minimal.

3. Final Answer:

   The edges array is not sorted by weight before processing -> Option D

4. Quick Check:

   Sorting edges first is essential [OK]

Hint: Always sort edges by weight before union-find steps [OK]

Common Mistakes:

• Skipping edge sorting
• Ignoring path compression (optional)
• Incorrect parent initialization

5. Given a graph with vertices 0 to 4 and edges:

[[0,1,10],[0,2,6],[0,3,5],[1,3,15],[2,3,4]]

Use Kruskal's Algorithm to find the total weight of the minimum spanning tree.

hard

A. 19

B. 20

C. 25

D. 15

Solution

1. Step 1: Sort edges by weight

   Sorted edges: [2,3,4], [0,3,5], [0,2,6], [0,1,10], [1,3,15]

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

   Pick [2,3,4], [0,3,5], [0,2,6] connect nodes 0,2,3. Then pick [0,1,10] to connect node 1. Total weight = 4+5+6+10 = 25. But this forms a cycle because 0,2,3 are connected. Instead, pick [2,3,4], [0,3,5], [0,1,10]. Then pick [0,2,6] creates cycle. So skip [0,2,6]. Next pick [1,3,15] is heavier and unnecessary. So MST edges: [2,3,4], [0,3,5], [0,1,10]. Total weight = 4+5+10=19.

3. Final Answer:

   19 -> Option A

4. Quick Check:

   Sum of MST edges weights = 19 [OK]

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

Common Mistakes:

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