Minimum Spanning Tree Kruskal's Algorithm in DSA C - 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 size grows.

Specifically, how does the number of edges and vertices affect the work done?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

// Kruskal's Algorithm main steps
sort(edges, edges + E, compare);
for (int i = 0; i < E; i++) {
  if (findSet(edges[i].u) != findSet(edges[i].v)) {
    unionSet(edges[i].u, edges[i].v);
    mst_weight += edges[i].weight;
  }
}

This code sorts all edges by weight and then adds edges to the MST if they connect different sets.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Sorting all edges and iterating through them.
• How many times: Sorting takes about E log E steps; the loop runs E times.
• Union-Find operations: Each union or find operation is almost constant time due to path compression.

How Execution Grows With Input

As the number of edges (E) and vertices (V) grow, sorting edges takes more time, and we check each edge once.

Input Size (E)	Approx. Operations
10	About 10 * log 10 ≈ 33 steps
100	About 100 * log 100 ≈ 664 steps
1000	About 1000 * log 1000 ≈ 9966 steps

Pattern observation: The work grows a bit faster than the number of edges because of sorting, but union-find operations stay very fast.

Final Time Complexity

Time Complexity: O(E log E)

This means the time grows mostly with sorting edges, and the rest of the steps add only small extra time.

Common Mistake

[X] Wrong: "The algorithm takes O(V^2) time because it checks all pairs of vertices."

[OK] Correct: Kruskal's algorithm works mainly with edges, not all vertex pairs, and sorting edges dominates the time, not checking all pairs.

Interview Connect

Understanding Kruskal's time complexity shows you can analyze sorting and efficient data structures like union-find, skills valuable in many algorithm problems.

Self-Check

"What if we used a different data structure for union-find without path compression? How would the time complexity change?"

Practice

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

easy

A. To sort the vertices of a graph

B. To find the shortest path between two nodes

C. To detect cycles in a graph

D. To find the minimum spanning tree 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 algorithms

   Unlike shortest path or cycle detection, Kruskal's focuses on building a minimum spanning tree.

3. Final Answer:

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

4. Quick Check:

   Kruskal's algorithm = Minimum Spanning Tree [OK]

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

Common Mistakes:

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

2. Which data structure is commonly used in Kruskal's algorithm to detect cycles efficiently?

easy

A. Disjoint Set (Union-Find)

B. Stack

C. Queue

D. Priority Queue

Solution

1. Step 1: Identify cycle detection method in Kruskal's

   Kruskal's algorithm uses a Disjoint Set (Union-Find) data structure to check if adding an edge creates a cycle by checking if two vertices belong to the same set.

2. Step 2: Compare other data structures

   Queue and stack do not help detect cycles efficiently here; priority queue is used for sorting edges but not cycle detection.

3. Final Answer:

   Disjoint Set (Union-Find) -> Option A

4. Quick Check:

   Cycle detection in Kruskal's = Union-Find [OK]

Hint: Use Union-Find to quickly check cycles in Kruskal's [OK]

Common Mistakes:

• Confusing priority queue with cycle detection
• Using stack or queue for cycle detection
• Not knowing Union-Find purpose

3. Consider the following edges of a graph with weights:
Edges = [(A-B, 1), (B-C, 4), (A-C, 3), (C-D, 2)]
What is the total weight of the minimum spanning tree found by Kruskal's algorithm?

medium

A. 7

B. 6

C. 8

D. 10

Solution

1. Step 1: Sort edges by weight

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

2. Step 2: Pick edges without cycles

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

3. Step 3: Calculate total weight

   Total weight = 1 + 2 + 3 = 6

4. Final Answer:

   6 -> Option B

5. Quick Check:

   Sum of MST edges = 6 [OK]

Hint: Sort edges, pick smallest without cycle until all connected [OK]

Common Mistakes:

• Adding all edges without checking cycles
• Choosing edges in wrong order
• Miscounting total weight

4. The following C code snippet is part of Kruskal's algorithm. What is the error?

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

void unionSet(int parent[], int x, int y) {
    int xset = findParent(parent, x);
    int yset = findParent(parent, y);
    parent[xset] = yset;
}

medium

A. unionSet should swap xset and yset

B. unionSet should not change parent array

C. Missing path compression in findParent

D. findParent should return void

Solution

1. Step 1: Analyze findParent function

   findParent recursively finds the root parent but does not update parent[i] to root, missing path compression optimization.

2. Step 2: Understand unionSet function

   unionSet correctly merges sets by updating parent of one root to another; no swap needed.

3. Final Answer:

   Missing path compression in findParent -> Option C

4. Quick Check:

   Path compression missing = findParent optimization [OK]

Hint: Add path compression to speed up findParent [OK]

Common Mistakes:

• Thinking unionSet must swap parents
• Believing findParent should return void
• Ignoring path compression importance

5. Given a graph with vertices {1, 2, 3, 4} and edges with weights:
(1-2, 5), (1-3, 3), (2-3, 4), (2-4, 2), (3-4, 7)
Using Kruskal's algorithm, which edges will be included in the minimum spanning tree?

hard

A. (2-4), (1-3), (2-3)

B. (1-3), (2-4), (1-2)

C. (2-4), (1-3), (1-2)

D. (1-2), (2-3), (3-4)

Solution

1. Step 1: Sort edges by weight

   Sorted edges: (2-4,2), (1-3,3), (2-3,4), (1-2,5), (3-4,7)

2. Step 2: Pick edges without cycles

   Pick (2-4,2), then (1-3,3). Next (2-3,4) connects 2 and 3 which are in different sets, so pick it. After these three edges, all vertices are connected and MST is complete.

3. Step 3: Verify options

   (2-4), (1-3), (2-3) matches edges (2-4), (1-3), (2-3). (2-4), (1-3), (1-2) includes (1-2) which is heavier than (2-3), so incorrect.

4. Final Answer:

   (2-4), (1-3), (2-3) -> Option A

5. Quick Check:

   Pick smallest edges connecting all vertices without cycle [OK]

Hint: Sort edges, pick smallest connecting new vertices until all connected [OK]

Common Mistakes:

• Choosing heavier edges over lighter ones
• Adding edges that create cycles
• Not stopping after n-1 edges