Minimum Spanning Tree Kruskal's Algorithm in DSA Typescript - Build from Scratch

Minimum Spanning Tree using Kruskal's Algorithm

📖 Scenario: You are working as a network engineer. You want to connect several computers with cables. Each cable has a cost. You want to connect all computers with the least total cost without any loops.

🎯 Goal: Build a program that finds the minimum total cost to connect all computers using Kruskal's algorithm.

📋 What You'll Learn

Create an array of edges with exact weights and nodes

Create a parent array for union-find structure

Implement the union-find functions: find and union

Sort edges by weight

Use Kruskal's algorithm to select edges without cycles

Print the total cost of the minimum spanning tree

💡 Why This Matters

🌍 Real World

Network design, connecting computers or cities with minimum cable or road cost.

💼 Career

Useful for software engineers working on network optimization, infrastructure planning, and graph algorithms.

1

Create the graph edges

Create an array called edges with these exact entries: {from: 0, to: 1, weight: 4}, {from: 0, to: 2, weight: 3}, {from: 1, to: 2, weight: 1}, {from: 1, to: 3, weight: 2}, {from: 2, to: 3, weight: 4}, {from: 3, to: 4, weight: 2}, {from: 4, to: 5, weight: 6}

DSA Typescript

// Create the edges array with given objects
// Your code here

Hint

Use an array of objects with keys from, to, and weight.

2

Create the parent array for union-find

Create an array called parent with length 6 where each element is its own index (0 to 5).

DSA Typescript

const edges = [
  {from: 0, to: 1, weight: 4},
  {from: 0, to: 2, weight: 3},
  {from: 1, to: 2, weight: 1},
  {from: 1, to: 3, weight: 2},
  {from: 2, to: 3, weight: 4},
  {from: 3, to: 4, weight: 2},
  {from: 4, to: 5, weight: 6}
]
// Create the parent array for union-find
// Your code here

Hint

Each node is its own parent initially.

3

Implement union-find and Kruskal's algorithm

Write the find function that returns the root parent of a node using path compression. Write the union function that connects two nodes if they have different parents. Sort the edges array by weight. Use a for loop to iterate over edges and add the edge's weight to totalCost if union returns true.

DSA Typescript

const edges = [
  {from: 0, to: 1, weight: 4},
  {from: 0, to: 2, weight: 3},
  {from: 1, to: 2, weight: 1},
  {from: 1, to: 3, weight: 2},
  {from: 2, to: 3, weight: 4},
  {from: 3, to: 4, weight: 2},
  {from: 4, to: 5, weight: 6}
]

const parent = [0, 1, 2, 3, 4, 5]

// Write find and union functions
// Sort edges by weight
// Use for loop to build MST and calculate totalCost
// Your code here

Hint

Use path compression in find. In union, connect roots if different. Sort edges by weight ascending. Add edge weight to totalCost only if union succeeds.

4

Print the total cost of the minimum spanning tree

Write console.log(totalCost) to display the total cost of the minimum spanning tree.

DSA Typescript

const edges = [
  {from: 0, to: 1, weight: 4},
  {from: 0, to: 2, weight: 3},
  {from: 1, to: 2, weight: 1},
  {from: 1, to: 3, weight: 2},
  {from: 2, to: 3, weight: 4},
  {from: 3, to: 4, weight: 2},
  {from: 4, to: 5, weight: 6}
]

const parent = [0, 1, 2, 3, 4, 5]

function find(x: number): number {
  if (parent[x] !== x) {
    parent[x] = find(parent[x])
  }
  return parent[x]
}

function union(a: number, b: number): boolean {
  const rootA = find(a)
  const rootB = find(b)
  if (rootA === rootB) return false
  parent[rootB] = rootA
  return true
}

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

let totalCost = 0
for (const edge of edges) {
  if (union(edge.from, edge.to)) {
    totalCost += edge.weight
  }
}

// Print the totalCost
// Your code here

Hint

The minimum total cost to connect all computers is 14.

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