This PR introduces a fully documented implementation of Johnson’s algorithm for computing shortest paths between all pairs of vertices in a sparse weighted directed graph in R.

Overview

The johnson_shortest_paths function computes shortest paths efficiently in graphs with positive and negative edge weights (but no negative cycles). It uses a combination of Bellman-Ford to reweight edges and Dijkstra’s algorithm for shortest path computation from each vertex. This ensures optimal handling of sparse graphs and negative edges without introducing negative cycles.

Features

• Computes shortest paths between all pairs of vertices
• Supports graphs with negative edge weights (no negative cycles allowed)
• Detects negative cycles and reports them clearly
• Handles disconnected components
• Returns a distance matrix for all vertex pairs
• Uses adjacency list representation consistent with other graph modules
• Includes demonstration with a sample graph, including negative edges

Complexity

• Time Complexity: O(V · E + V · (E log V)) with a binary heap priority queue
• Space Complexity: O(V + E)

Demonstration

Run the included demo code to see Johnson’s algorithm in action on a 5-vertex graph with negative edges but no negative cycles. The output is a distance matrix representing shortest paths between all vertices.

Summary

This implementation expands the R graph algorithms module by adding a core algorithm for efficient all-pairs shortest paths in sparse graphs, suitable for network analysis, routing, and optimization.