Add Dijkstra's shortest path algorithm implementation #[HACKTOBERFEST 2025]

[piyushkumar0707]

🚀 Overview

This PR implements Dijkstra's shortest path algorithm for finding optimal paths in weighted graphs - a cornerstone algorithm in graph theory and network optimization.

✨ Features

• ✅ Complete Dijkstra implementation - Handles weighted, non-negative graphs
• ✅ Priority queue optimization - Efficient vertex selection
• ✅ Path reconstruction - Full shortest paths, not just distances
• ✅ Multiple utility functions - Single-target and all-paths variants
• ✅ Comprehensive examples - Complex weighted graphs with verification
• ✅ Educational documentation - Step-by-step algorithm explanation

🎯 Why This Matters

Dijkstra's algorithm is crucial for:

• GPS navigation systems - Finding shortest routes
• Network routing protocols - Internet packet routing
• Game AI pathfinding - Character movement optimization
• Social network analysis - Finding influence paths
• Operations research - Supply chain optimization
• Flight scheduling - Cheapest flight connections

📚 Implementation Details

• Time Complexity: O((V + E) log V) with binary heap
• Space Complexity: O(V) for distance and previous arrays
• Algorithm: Greedy approach with relaxation
• Key Functions:
  • dijkstra_shortest_path() - Core algorithm
  • get_shortest_path() - Path reconstruction
  • get_all_shortest_paths() - All paths from source

🔧 Technical Features

• Priority Queue: Custom implementation for educational purposes
• Relaxation Step: Proper distance updates with predecessor tracking
• Path Reconstruction: Backtracking using predecessor array
• Input Validation: Handles disconnected vertices gracefully

🧪 Testing

Extensive examples including:

• Simple weighted graphs with manual verification
• Complex networks with multiple optimal paths
• Edge cases (unreachable vertices, self-loops)
• Performance comparisons with different graph sizes

📁 Files Added

• graph_algorithms/dijkstra_shortest_path.r - Complete implementation
• Updated DIRECTORY.md with Dijkstra entry

🏆 Educational Value

Includes detailed comments explaining:

• Priority queue operations
• Relaxation process
• Path reconstruction technique
• Time complexity analysis

🎃 Hacktoberfest 2025

This completes the fundamental graph algorithms trilogy (DFS, BFS, Dijkstra) providing comprehensive graph algorithm coverage for the repository.

[Copilot]

Pull Request Overview

Implements Dijkstra's shortest path algorithm with path reconstruction and utility helpers, plus documentation-style inline examples.

• Adds core algorithm (priority-queue based), path reconstruction, and all-paths helper.
• Introduces inline example / demo executions within the same source file.
• Updates DIRECTORY.md to list the new algorithm.

Reviewed Changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated 5 comments.

File	Description
graph_algorithms/dijkstra_shortest_path.r	Adds Dijkstra implementation, priority queue helpers, path reconstruction, and embedded runnable examples.
DIRECTORY.md	Registers the new Dijkstra algorithm in the index.

---

_{Tip: Customize your code reviews with copilot-instructions.md. Create the file or learn how to get started.}

[siriak]

Please check comments

[piyushkumar0707]

@siriak all conflicts resolved