Efficient computations of discrete cubical homology

TL;DR

This paper introduces an optimized algorithm for computing discrete cubical homology of graphs over finite fields, enhancing efficiency through innovative generation, quotienting, and preprocessing techniques.

Contribution

The paper presents a novel, faster algorithm for discrete cubical homology computation that improves upon existing methods with key insights into generation, automorphism quotienting, and preprocessing.

Findings

Significantly faster computation times compared to previous methods

Reduced vector space dimensions via automorphism quotienting

Effective preprocessing of graphs for homology calculation

Abstract

We present a fast algorithm for computing discrete cubical homology of graphs over finite fields with an appropriate characteristic. This algorithm improves on several computational steps compared to constructions in the existing literature, with the key insights including: a faster way to generate all singular cubes, reducing the dimensions of vector spaces in the chain complex by taking a quotient over automorphisms of the cube, and preprocessing graphs using the axiomatic treatment of discrete cubical homology.