Kruskal’s Algorithm is a popular algorithm in graph theory used to find the Minimum Spanning Tree (MST) of a connected, weighted graph. MST is a subset of the graph’s edges that connects all the vertices with the minimum total edge weight.

Understanding Kruskal’s Algorithm

Graph Representation

The first step in implementing Kruskal’s Algorithm is to represent the graph. We can use either an adjacency list or an adjacency matrix to store the graph’s edges and their weights.

Algorithm Steps

1. Sort the edges of the graph in non-decreasing order based on their weights.
2. Create an empty set to store the MST.
3. Iterate through each edge in the sorted order.
   • If adding the current edge to the MST doesn’t form a cycle, add it to the MST set.
   • Otherwise, ignore the edge.
4. Return the MST set.

Example Usage

Let’s consider the following undirected graph with its corresponding weights:

        4
   (0)-------(1)
    |         |
   2|       3 |9
    |         |
   (2)-------(3)
      \   6
       \ |
        (4)
          \
           1

Using Kruskal’s Algorithm, we can find the Minimum Spanning Tree for this graph. Here are the steps:

1. Sort the edges based on weights: {(4,2,6), (1,0,4), (0,3,9), (0,1,2), (2,3,3)}.
2. Create an empty set for the MST: mst = {}.
3. Iterate over each edge:
   • (4,2,6): Add it to mst.
   • (1,0,4): Add it to mst.
   • (0,3,9): Add it to mst.
   • (0,1,2): This edge forms a cycle, so ignore it.
   • (2,3,3): Add it to mst.

After the algorithm completes, the Minimum Spanning Tree for the given graph will consist of the edges {(4,2,6), (1,0,4), (0,3,9), (2,3,3)}.

Complexity Analysis

The time complexity of Kruskal’s Algorithm is primarily dominated by the sorting step, which takes O(E log E) time, where E is the number of edges in the graph. The union-find operations to detect cycles take nearly constant time due to path compression and union by rank optimizations.

The overall time complexity of Kruskal’s Algorithm is O(E log E) or O(E log V), depending on the way the graph is represented (E: number of edges, V: number of vertices).

Conclusion

Kruskal’s Algorithm is an efficient way to find the Minimum Spanning Tree in a connected, weighted graph. By considering the edges in a sorted order and ensuring that they don’t form cycles, we can obtain the MST that minimizes the total weight. Understanding this algorithm is essential for solving various graph theory problems. #KruskalAlgorithm #MinimumSpanningTree