● Covered by 1 source · 1 reportMedium impact1 neutral
New C-HD Algorithm Improves Shortest Path Calculation for Dense Graphs
Key points
- C-HD algorithm calculates exact shortest paths.
- Achieves O(V + E log log V) complexity.
- Outperforms Dijkstra for E between V log log V and V log V.
- Algorithm formally verified using Lean.
New Algorithm for Shortest Paths
A new algorithm, C-HD, has been proposed for finding exact shortest-path distances in directed graphs. The algorithm is designed for graphs with non-negative real edge weights and was developed and formally verified using the Lean proof assistant.
The C-HD algorithm handles local searches and unproductive edges efficiently. It uses priority comparisons and maintains local invariants through careful edge deletion and bounded local search.
Improved Time Complexity
C-HD achieves a time complexity of O(V + E log log V), where V is the number of vertices and E is the number of edges. This bound is certified for graphs where E is within the range of V log log V to V log V.
This performance includes overheads for memory allocation, edge sorting, input graph reading, and result output. The algorithm's improved bound stems from its ability to reduce repeated search and data-structure operations.
Comparison with Existing Algorithms
Dijkstra's shortest path algorithm typically runs in O(E + V log V) time using a Fibonacci heap. While other deterministic algorithms achieve O(E + V log V) or O(E + V log log V) for specific conditions, C-HD offers a better bound in the parameter space where E is between V log log V and V log V.
This new algorithm provides an alternative for scenarios where the graph density falls within its certified range, potentially offering faster computations than traditional methods.
Formal Verification with Lean
The development of C-HD involved formal verification using the Lean proof assistant. This process, which took approximately 15 hours and 733 messages, ensured the correctness and efficiency of the proposed algorithm. Formal verification provides a high degree of assurance regarding the algorithm's reliability.
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Primary sources
GitHub spicylemonade/c-hd-proof GitHub spicylemonade/c-hd-proof. arXiv 2504.17033 arXiv 2602.07868Reporting from
A new algorithm, C-HD, has been developed to calculate exact shortest path distances in directed graphs with non-negative real weights. C-HD achieves a time complexity of O(V + E log log V) for graphs where E is between V log log V and V log V, outperforming Dijkstra's algorithm in this specific range. The algorithm was formally verified using Lean and reduces repeated search and data-structure work.