Add Breadth-First Search (BFS) algorithm implementation #[HACKTOBERFEST 2025]

[piyushkumar0707]

🚀 Overview

This PR adds a comprehensive implementation of the Breadth-First Search (BFS) algorithm with advanced features including shortest path finding and distance-based vertex discovery.

✨ Features

• ✅ Queue-based BFS traversal - Level-by-level graph exploration
• ✅ Shortest path finding - Find shortest unweighted path between vertices
• ✅ Distance-based search - Find all vertices at specific distance
• ✅ Path reconstruction - Complete path from source to target
• ✅ Comprehensive documentation - Time O(V+E) and space O(V) complexity
• ✅ Multiple utility functions - Various BFS applications in one file

🎯 Why This Matters

BFS is essential for:

• Shortest path in unweighted graphs
• Level-order traversal of trees
• Connected components analysis
• Social network analysis (degrees of separation)
• Web crawling algorithms
• GPS navigation (unweighted scenarios)

📚 Implementation Details

• Time Complexity: O(V + E) for traversal, path finding
• Space Complexity: O(V) for queue and visited tracking
• Algorithm: Queue-based level-by-level exploration
• Special Features:
  • bfs_shortest_path() - Returns path and distance
  • bfs_vertices_at_distance() - Finds vertices at exact distance

🧪 Testing

Comprehensive examples including:

• Basic BFS traversal from different starting points
• Shortest path finding with path reconstruction
• Distance-based vertex discovery
• Complex connected graphs with multiple components

📁 Files Added

• graph_algorithms/breadth_first_search.r - Complete BFS implementation
• Updated DIRECTORY.md with BFS entry

🔗 Complements

This pairs perfectly with DFS to provide complete graph traversal coverage for the repository.

🎃 Hacktoberfest 2025

Adds another fundamental algorithm with practical applications, enhancing the repository's educational value for computer science students.

[Copilot]

Pull Request Overview

Adds a new Breadth-First Search (BFS) implementation with shortest path and distance-based utilities, plus an index entry in DIRECTORY.md.

• Implements three BFS-related functions (traversal, shortest path, vertices at distance).
• Adds embedded example/demo code executing on load.
• Updates directory listing with a Graph Algorithms section.

Reviewed Changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated 12 comments.

File	Description
graph_algorithms/breadth_first_search.r	Adds BFS traversal, shortest path, and distance query functions plus inline executable examples.
DIRECTORY.md	Adds Graph Algorithms section pointing to new BFS implementation.

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[siriak]

Please check the comments

[piyushkumar0707]

@siriak sir all conflicts has been resolved