# 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.
- Title: C-HD Algorithm Achieves Faster Shortest Path Computation
- Summary: 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…
- Keywords: c-hd, shortest path, dijkstra, lean, algorithm, graph theory, computational complexity, non-negative weights, directed graph, runtime bound, mathematical proof, formal verification
- Source: Vals — https://vals.ai/blogs/faster-shortest-path-algorithm
- Published: 2026-09-20T00:00:00+00:00
- Read time: 1 min
- Topics: c-hd, shortest path, dijkstra, lean, algorithm, graph theory, computational complexity, non-negative weights, directed graph, runtime bound, mathematical proof, formal verification
## 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.
## Key takeaway

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.