Minimum Spanning Tree Kruskal's Algorithm in DSA Typescript - Time & Space Complexity

Time Complexity: Minimum Spanning Tree Kruskal's Algorithm

O(E log E)

Understanding Time Complexity

We want to understand how the time needed to find a minimum spanning tree using Kruskal's algorithm changes as the graph grows.

Specifically, how the number of edges and vertices affects the work done.

Scenario Under Consideration

Analyze the time complexity of the following Kruskal's algorithm snippet.

function kruskal(edges: [number, number, number][], verticesCount: number) {
  edges.sort((a, b) => a[2] - b[2]);
  const parent = Array(verticesCount).fill(0).map((_, i) => i);
  const rank = Array(verticesCount).fill(0);

  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;
    if (rank[rootA] < rank[rootB]) parent[rootA] = rootB;
    else if (rank[rootB] < rank[rootA]) parent[rootB] = rootA;
    else {
      parent[rootB] = rootA;
      rank[rootA]++;
    }
    return true;
  }

  const mst = [];
  for (const [u, v, w] of edges) {
    if (union(u, v)) mst.push([u, v, w]);
  }
  return mst;
}
    

This code sorts edges by weight, then adds edges to the MST if they connect different sets, using union-find to avoid cycles.

Identify Repeating Operations

Look for loops and repeated calls that take most time.

• Primary operation: Sorting all edges by weight.
• How many times: Sorting runs once on all edges, which is E edges.
• Secondary operation: The loop over all edges to try union operations.
• Union-Find operations: Each union or find runs almost constant time, but called up to E times.

How Execution Grows With Input

As the number of edges (E) and vertices (V) grow, the sorting step dominates.

Input Size (E, V)	Approx. Operations
10 edges, 5 vertices	About 10 log 10 = 33 operations for sorting + 10 union-find calls
100 edges, 50 vertices	About 100 log 100 = 664 operations for sorting + 100 union-find calls
1000 edges, 500 vertices	About 1000 log 1000 = 9965 operations for sorting + 1000 union-find calls

Pattern observation: The sorting grows faster than the union-find calls, roughly proportional to E times log E.

Final Time Complexity

Time Complexity: O(E log E)

This means the time grows mostly with sorting the edges, which depends on the number of edges times the logarithm of that number.

Common Mistake

[X] Wrong: "The union-find operations take as much time as sorting."

[OK] Correct: Union-find operations are almost constant time due to path compression and ranking, so sorting dominates the total time.

Interview Connect

Understanding Kruskal's time complexity shows you can analyze algorithms that combine sorting and efficient data structures, a key skill in many coding challenges.

Self-Check

"What if the edges were already sorted? How would that affect the time complexity?"

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