Kruskal's Algorithm Calculator

🧠 Deep Dive into Kruskal's Algorithm

Welcome to the ultimate guide on Kruskal's algorithm. Whether you're a computer science student, a competitive programmer, or just curious about graph theory, this comprehensive resource will provide everything you need to know about finding the Minimum Spanning Tree (MST).

❓ What is Kruskal's Algorithm?

Kruskal's algorithm is a famous greedy algorithm in graph theory used to find the Minimum Spanning Tree (MST) for a connected, undirected graph with weighted edges. A Minimum Spanning Tree is a subset of the edges of the graph that connects all the vertices together, without any cycles, and with the minimum possible total edge weight.

Think of it like this: Imagine you need to connect several cities with a fiber optic cable network. You want to use the least amount of cable possible while ensuring every city is connected to the network. Kruskal's algorithm provides an optimal solution to this problem.

Core Principles:

• 🌳 Spanning Tree: It creates a 'tree' (a graph with no cycles) that 'spans' (connects) all vertices.
• ⚖️ Minimum Weight: Among all possible spanning trees, it finds the one with the smallest sum of edge weights.
• 💡 Greedy Approach: At each step, it makes the locally optimal choice by picking the smallest available edge, hoping this will lead to a globally optimal solution (which, in this case, it does!).

⚙️ How Does Kruskal's Algorithm Work? A Step-by-Step Guide

The beauty of Kruskal's algorithm lies in its simplicity. It builds the MST by iteratively adding the best possible edge. Here's a detailed breakdown of the steps involved:

1. 1️⃣ Sort All Edges: The first and most crucial step is to create a list of all edges in the graph and sort them in non-decreasing order of their weights.
2. 2️⃣ Initialize the MST: Create an empty set or list, which will eventually hold the edges of our Minimum Spanning Tree.
3. 3️⃣ Initialize Disjoint Sets: Treat each vertex in the graph as a separate, independent component or 'set'. This is where the Disjoint Set Union (DSU) or Union-Find data structure comes into play. It's incredibly efficient at keeping track of which vertices are connected.
4. 4️⃣ Iterate and Build: Go through the sorted list of edges one by one, from the smallest weight to the largest. For each edge (u, v):
   • Check if vertices 'u' and 'v' are already in the same component (i.e., if adding this edge would form a cycle). We do this using the `find` operation of our DSU data structure.
   • If `find(u)` is not equal to `find(v)`, it means they are in different components, and adding this edge will not create a cycle.
   • In this case, add the edge (u, v) to our MST.
   • Then, merge the two components of 'u' and 'v' into a single component using the `union` operation of the DSU.
   • If `find(u)` is equal to `find(v)`, do nothing and discard this edge, as adding it would create a cycle.
5. 5️⃣ Termination: Continue this process until the MST has (V-1) edges, where V is the number of vertices. At this point, all vertices will be connected, and we will have our Minimum Spanning Tree.

⏳ Kruskal's Algorithm Time Complexity Explained

Understanding the runtime or time complexity is vital for evaluating an algorithm's efficiency. For Kruskal's algorithm, the complexity is dominated by two main operations:

• Sorting the Edges: If there are 'E' edges, sorting them using an efficient comparison-based sorting algorithm (like Merge Sort or Heap Sort) takes O(E log E) time.
• Union-Find Operations: We iterate through all 'E' edges. For each edge, we perform two `find` operations and at most one `union` operation. A highly optimized DSU data structure with path compression and union by rank/size makes these operations nearly constant time. The complexity is technically O(α(V)), where α is the Inverse Ackermann function, which grows so slowly that it's considered effectively constant for all practical purposes. Therefore, the total time for all DSU operations is approximately O(E α(V)).

Combining these, the overall time complexity of Kruskal's algorithm is O(E log E + E α(V)). Since `log E` is typically larger than `α(V)`, the complexity is simplified to O(E log E). If the edges are already sorted, or can be sorted in linear time (e.g., using Radix Sort for integer weights), the complexity can be improved to O(E α(V)).

🔍 Kruskal's Algorithm Example (Step-by-Step)

Let's walk through a concrete example. Consider a graph with 4 vertices (A, B, C, D) and the following edges:

• A - B: Weight 10
• A - C: Weight 6
• A - D: Weight 5
• B - D: Weight 15
• C - D: Weight 4

Step 1: Sort Edges by Weight

1. C - D: 4
2. A - D: 5
3. A - C: 6
4. A - B: 10
5. B - D: 15

Step 2: Initialize

• MST = {} (empty)
• Components = {A}, {B}, {C}, {D}

Step 3: Iterate and Build

1. Edge (C, D) weight 4: `find(C) != find(D)`. No cycle. ✅ Add to MST.
   • MST = {(C, D)}
   • Union(C, D). Components = {A}, {B}, {C, D}
2. Edge (A, D) weight 5: `find(A) != find(D)`. No cycle. ✅ Add to MST.
   • MST = {(C, D), (A, D)}
   • Union(A, D). Components = {B}, {A, C, D}
3. Edge (A, C) weight 6: `find(A) == find(C)`. They are in the same component {A, C, D}. ❌ This would form a cycle (A-D-C). Discard.
4. Edge (A, B) weight 10: `find(A) != find(B)`. No cycle. ✅ Add to MST.
   • MST = {(C, D), (A, D), (A, B)}
   • Union(A, B). Components = {A, B, C, D}

Step 4: Terminate

We now have 3 edges (V-1), and all vertices are in one component. The algorithm stops. The resulting Minimum Spanning Tree consists of edges (C,D), (A,D), and (A,B), with a total weight of 4 + 5 + 10 = 19.

🆚 Kruskal's Algorithm vs. Prim's Algorithm

Kruskal's and Prim's are the two most famous algorithms for finding the MST. While they always produce an MST with the same total weight (though the set of edges might differ if multiple MSTs exist), their approaches are fundamentally different.

🌍 Applications of Kruskal's Algorithm

Kruskal's algorithm is not just a theoretical concept; it has numerous real-world applications:

• 🔌 Network Design: Laying electrical grids, water pipelines, or fiber optic cables to connect multiple locations with minimum cost.
• 🗺️ Transportation Networks: Designing road or railway networks to connect cities with the minimum total length.
• 🔬 Bioinformatics: Analyzing gene clusters and relationships in biological data.
• 🖼️ Image Processing: Used in algorithms for image segmentation and clustering.
• 🤖 Circuit Design: Finding the most efficient way to wire components on a circuit board.