Minimum Spanning Tree Kruskal's Algorithm in DSA Java - 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 the minimum spanning tree using Kruskal's algorithm changes as the graph 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 to find MST
class Edge implements Comparable<Edge> {
  int src, dest, weight;
  public int compareTo(Edge other) {
    return this.weight - other.weight;
  }
}

void kruskalMST(List<Edge> edges, int V) {
  Collections.sort(edges); // sort edges by weight
  DisjointSet ds = new DisjointSet(V);
  for (Edge e : edges) {
    if (ds.find(e.src) != ds.find(e.dest)) {
      ds.union(e.src, e.dest);
      // add edge to MST
    }
  }
}
    

This code sorts all edges by weight, then adds edges to the MST if they connect different sets, using a disjoint set data structure.

Identify Repeating Operations

• Primary operation: Sorting all edges and iterating through each edge once.
• How many times: Sorting takes time based on number of edges E; the loop runs once per edge (E times).
• Disjoint set operations: Each union/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, the sorting step dominates the time.

Input Size (V, E)	Approx. Operations
V=10, E=15	Sorting ~ 15 log 15 ≈ 60 operations; loop 15 times
V=100, E=500	Sorting ~ 500 log 500 ≈ 4500 operations; loop 500 times
V=1000, E=5000	Sorting ~ 5000 log 5000 ≈ 65000 operations; loop 5000 times

Pattern observation: The sorting step grows faster than the loop, so total time grows roughly proportional to E log E.

Final Time Complexity

Time Complexity: O(E log E)

This means the time grows mostly with sorting all edges, which depends on the number of edges and grows a bit faster than just the number of edges.

Common Mistake

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

[OK] Correct: Kruskal's algorithm works mainly with edges, not all vertex pairs. It sorts edges and uses efficient union-find, so time depends on edges, not all vertex pairs.

Interview Connect

Understanding how Kruskal's algorithm scales helps you explain your approach clearly and shows you know how to handle graphs efficiently.

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 detect cycles in a graph

B. To find the shortest path between two nodes

C. To find the minimum spanning tree of 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.

3. Final Answer:

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

4. Quick Check:

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

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

Common Mistakes:

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

2. Which Java data structure is commonly used to efficiently check and union sets in Kruskal's algorithm?

easy

A. Disjoint Set (Union-Find)

B. PriorityQueue

C. ArrayList

D. HashMap

Solution

1. Step 1: Identify the need for set operations

   Kruskal's algorithm requires checking if two vertices belong to the same set and merging sets without cycles.

2. Step 2: Match data structure to operations

   Disjoint Set (Union-Find) supports fast union and find operations, ideal for this purpose.

3. Final Answer:

   Disjoint Set (Union-Find) -> Option A

4. Quick Check:

   Union-Find = fast set checks and merges [OK]

Hint: Use Union-Find to detect cycles fast in Kruskal [OK]

Common Mistakes:

• Using ArrayList for set operations
• Confusing PriorityQueue with Union-Find
• Using HashMap which is not efficient here

3. What is the output MST edges after running Kruskal's algorithm on this graph?

Vertices: 1, 2, 3, 4
Edges with weights:
1-2: 1
1-3: 3
2-3: 2
3-4: 4
2-4: 5

medium

A. [1-2, 2-3, 3-4]

B. [1-3, 2-3, 3-4]

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

D. [1-2, 2-4, 3-4]

Solution

1. Step 1: Sort edges by weight

   Edges sorted: 1-2(1), 2-3(2), 1-3(3), 3-4(4), 2-4(5)

2. Step 2: Pick edges without cycles

   Pick 1-2, then 2-3 (no cycle), then 1-3 creates cycle so skip, pick 3-4 (no cycle), skip 2-4 (cycle)

3. Final Answer:

   [1-2, 2-3, 3-4] -> Option A

4. Quick Check:

   Edges chosen by increasing weight without cycles [OK]

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

Common Mistakes:

• Picking edges that form cycles
• Not sorting edges by weight
• Choosing heavier edges first

4. Identify the error in this Java snippet for Kruskal's algorithm:

for (Edge e : edges) {
  if (findSet(e.u) != findSet(e.v)) {
    union(e.u, e.v);
    mst.add(e);
  }
}
Collections.sort(edges);

medium

A. Union should be called before findSet

B. Edges should be sorted before the loop

C. mst.add(e) should be outside the if condition

D. The loop should iterate over mst, not edges

Solution

1. Step 1: Check order of sorting and iteration

   Kruskal's requires edges sorted by weight before processing to pick smallest edges first.

2. Step 2: Identify mistake in code order

   Sorting is done after the loop, so edges are not processed in correct order.

3. Final Answer:

   Edges should be sorted before the loop -> Option B

4. Quick Check:

   Sort edges first, then pick edges [OK]

Hint: Sort edges before loop to pick smallest first [OK]

Common Mistakes:

• Sorting edges after processing
• Calling union before checking sets
• Adding edges outside condition

5. Given a graph with edges:

Edges: (A-B, 4), (A-C, 3), (B-C, 2), (B-D, 5), (C-D, 7), (C-E, 8), (D-E, 6)

Using Kruskal's algorithm, what is the total weight of the minimum spanning tree?

hard

A. 18

B. 14

C. 20

D. 16

Solution

1. Step 1: Sort edges by weight

   Sorted edges: B-C(2), A-C(3), A-B(4), B-D(5), D-E(6), C-D(7), C-E(8)

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

   Pick B-C(2), A-C(3), A-B(4) forms cycle so skip, pick B-D(5), pick D-E(6). Total edges chosen: B-C, A-C, B-D, D-E.

3. Step 3: Calculate total weight

   2 + 3 + 5 + 6 = 16. But A-B(4) skipped due to cycle, so MST edges are B-C, A-C, B-D, D-E.

4. Step 4: Re-check cycle with A-B(4)

   Actually, A-B(4) creates cycle with A-C and B-C, so skip it. MST weight is 2+3+5+6=16.

5. Final Answer:

   16 -> Option D

6. Quick Check:

   Sum of chosen edges weights = 16 [OK]

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

Common Mistakes:

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