Introduction

Kruskal's Algorithm: Kruskal's Algorithm is a greedy algorithm used to find the Minimum Spanning Tree (MST) of a connected, weighted graph. It ensures that the total weight of the MST is minimized.
This algorithm is widely used in network design, such as designing least-cost networks.

How does it work?

• Sort all the edges of the graph in non-decreasing order of their weights.
• Initialize an empty set to store the edges of the MST.
• Iterate through the sorted edges and add an edge to the MST if it does not form a cycle with the edges already in the MST.
• Use a union-find data structure to efficiently check for cycles.
• Repeat the process until the MST contains exactly (V - 1) edges, where V is the number of vertices.

Important Observations

• The graph must be connected for the algorithm to work correctly.
• The algorithm guarantees the MST for weighted graphs.
• It uses a greedy approach to select the smallest edge at each step.

Key Characteristics:

• Greedy algorithm for MST calculation.
• Efficient for sparse graphs.
• Works for both directed and undirected graphs.

Advantages:

• Simple and easy to implement.
• Efficient for graphs with fewer edges.
• Uses union-find for cycle detection, which is efficient.

Disadvantages:

• Sorting edges can be computationally expensive for dense graphs.
• Not suitable for dynamic graphs where edges are frequently added or removed.

Time Complexity:

• Using union-find with path compression: O(E log V), where E is the number of edges and V is the number of vertices.