PR: Prim's Algorithm for Minimum Spanning Tree (MST)

This PR introduces a fully documented implementation of Prim's Algorithm in R, designed to compute the Minimum Spanning Tree (MST) of a connected, weighted, undirected graph. The algorithm efficiently grows the MST by repeatedly adding the minimum-weight edge connecting the already included vertices to a new vertex.

Overview

Prim's algorithm operates as follows:

• Start from an arbitrary vertex.
• Repeatedly select the smallest edge connecting a vertex in the MST to a vertex outside the MST.
• Add the selected edge and vertex to the MST.
• Continue until all vertices are included.

The implementation supports both adjacency matrix and edge list representations, includes input validation, and provides formatted MST output.

Features

• Computes the MST of connected, weighted, undirected graphs.
• Handles edge cases like disconnected graphs with appropriate warnings.
• Outputs MST edges, total weight, and parent array.
• Includes helper functions for printing MST and constructing adjacency matrices from edge lists.
• Multiple examples provided: basic graphs, complete graphs, edge-list graphs, and disconnected graphs.
• Comparison notes with Kruskal's algorithm for educational purposes.

Complexity

• Time Complexity:
  • O(V²) with adjacency matrix (simple implementation)
  • O((V + E) log V) with priority queue (optimized, not implemented here)
• Space Complexity: O(V) for key, parent, and visited arrays

Applications

• Network design (telecommunications, computer networks)
• Circuit design and electrical grids
• Clustering algorithms in machine learning
• Transportation and logistics planning
• Approximation algorithms for NP-hard problems

Demonstration

Run the included examples to see Prim's Algorithm in action:

# Example: Basic 5-vertex graph
graph <- matrix(c(
  0, 2, 0, 6, 0,
  2, 0, 3, 8, 5,
  0, 3, 0, 0, 7,
  6, 8, 0, 0, 9,
  0, 5, 7, 9, 0
), nrow = 5, byrow = TRUE)

mst <- prim_mst(graph)
print_mst(mst, graph)