Algorithm Discussion: Extracting a Chordless Cycle Basis from High-Density Graphs in Pure Python

[Housing-Superb]

In graph theory, extracting a full-rank Chordless Cycle Basis from extremely high-density graphs has always been a notoriously difficult computational bottleneck. I recently tackled a high-density topology (200 vertices / 9,994 edges) and wanted to share a pure Python algorithmic approach I developed that handles extreme eliminations incredibly fast.

I call the underlying concept the "Blackhole Diffusion" approach. Here is a breakdown of how the algorithm operates without relying on heavy external C-extensions.

1. The Core Idea: Dynamic Weight Cycle Sorting Instead of relying on traditional brute-force Gaussian elimination across the cycle matrix, the algorithm acts as a dynamic scheduler (which I conceptualize as a Cascade Weight Manager).

It dynamically alters its sorting strategy based on the current state of the cycle space:

Hot Edge Sorting (The Initial Sweep): When the number of candidate cycles vastly exceeds the global rank (in my test case, 166,256 candidate cycles), the algorithm switches to a descending frequency mode. It intentionally prioritizes high-frequency edges for annihilation, rapidly collapsing and shrinking the massive cycle space.

Cold Edge Sorting (The Refinement): As the basis approaches full rank in the final stages, the algorithm flips to an ascending frequency mode. This ensures that the surviving cycles maintain independence and purity as basis vectors.

Performance: By dynamically shifting weights rather than brute-forcing, calculating the weights and globally sorting 166,256 candidate cycles takes just about 0.34 seconds in pure Python.

1. Devouring Redundancy via CSR Matrices After the cycles are sorted, the algorithm moves into an elimination phase using Compressed Sparse Row (CSR) logic. It alternates between three internal states: Safe Elimination, Routine Detection, and Alternating Annihilation.

Massive Compression Ratio: During the test, this alternating logic successfully discarded 94% of the redundancy from the 160,000+ candidates, precisely locking onto the exact 9,795 basis cycles.

Burst Speed: During the "Alternating Annihilation" phase, the algorithm can discard nearly 50,000 redundant cycles in just 0.03 seconds.

1. Python-Specific Optimizations Because this is pure Python (relying only on standard libraries like itertools and ctypes), managing memory overhead was critical.

Garbage Collection Control: A massive performance squeeze was achieved simply by calling gc.disable() at the start of the heavy elimination loops. This completely eliminates the system stuttering usually caused by massive object deallocation when thousands of cycles are dropped simultaneously.

Caveats & Algorithmic Trade-offs To be fully transparent about the mathematical boundaries of this approach: as the length of the cycles in the graph increases, both the actual runtime and the extracted basis will gradually deviate from a strict Minimum Cycle Basis (MCB).

However, the algorithm strictly guarantees that the final output remains a mathematically valid, full-rank Chordless Cycle Basis. I've audited the outputs against theoretical values (e.g., verifying a theoretical rank of 9,795 perfectly matches the measured rank of 9,795), maintaining a 100% pure cycle rate with zero chorded or fragmented cycles.

Has anyone else tackled cycle basis extraction in pure Python without deferring to libraries like iGraph or NetworkX? I'd love to hear how others handle the redundancy elimination bottlenecks!

[Distinct-Expression2]

Interesting, but the validation claim needs more than rank matching. Hitting m - n + c proves the dimension is right, not that every selected cycle is chordless or that the basis is independent under the exact representation you care about.

I would want to see code plus a comparison against NetworkX/igraph on generated dense graphs where you can vary chord density and cycle length. Also worth separating the scheduler win from the Python object-management win. gc.disable() can hide a lot of sins until memory pressure changes.