From Vals · · 1m 50s
C-HD Algorithm Achieves Faster Shortest Path Computation
New algorithm finds exact shortest paths in directed graphs with non-negative weights faster than Dijkstra's method for specific input sizes.
In brief
New algorithm finds exact shortest paths in directed graphs with non-negative weights faster than Dijkstra's method for specific input sizes. C-HD algorithm achieves O(n log^{11/12} n) runtime for graphs with m ≈ n log^{3/4} n edges, improving upon Dijkstra's O(n log n) for large inputs. Originally reported by Vals.
Problem Definition
Shortest paths requires finding minimum weight paths between vertices in directed graphs with non-negative real weights.
Dijkstra's Limitation
Dijkstra's algorithm runs in O(m + n log n) time but has worse performance for graphs with m ≥ n edges.
C-HD Algorithm
C-HD algorithm uses local search with unproductive edges to achieve better runtime bounds for specific graph densities.
Certified Runtime
C-HD achieves O(n + m + m log(2 + m/(n+1)) + m^{1/3}(n log(n+2))^{2/3}) runtime within certified range.
Certified Density
Certified range requires m ≤ n floor(floor(log_2 n)^{3/4}) edges for the algorithm to be optimal.
Theoretical Improvement
For m ≈ n log^{3/4} n edges, C-HD runs in O(n log^{11/12} n) time versus Dijkstra's O(n log n).
Performance Ratio
The ratio of leading terms for n = 2^1000 is 1000^{1/12} ≈ 1.78, showing polylogarithmic improvement.
Practical Limitations
The algorithm's theoretical improvement doesn't translate to practical speedups for real-world graphs.
Formal Verification
Lean proof establishes runtime bound and shows strict asymptotic improvement over n log n for specific graph densities.
Agent Collaboration
10 Claude Opus 5.5 agents worked together on message board for 15 hours to develop the algorithm.
Implementation Details
Algorithm uses sorted outgoing-edge lists and falls back to Bellman, Ford for small inputs outside certified range.
Proof Validation
Lean Comparator tool verified proof uses permitted axioms and matches Lean's kernel acceptance criteria.
Real-World Application
C-HD algorithm provides better theoretical performance for specific graph densities but practical speedup remains unproven.
What it means
C-HD algorithm achieves O(n log^{11/12} n) runtime for graphs with m ≈ n log^{3/4} n edges, improving upon Dijkstra's O(n log n) for large inputs.
