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Efficient computations of discrete cubical homology

Chris Kapulkin, Nathan Kershaw

TL;DR

This work tackles the computational intractability of discrete cubical homology for graphs by combining mathematical and programming advances. The authors introduce a faster graph-map generation technique, leverage a quotient by the hyperoctahedral group to reduce module dimensions, and apply discrete-homotopy preprocessing to shrink graphs before computation. They also optimize matrix construction via degeneracy tracking, precomputed faces, and coordinate dictionaries, and demonstrate substantial speedups—enabling previously intractable cases such as high-dimensional homology and larger graphs, especially with parallelization. The result is the fastest known algorithm for discrete cubical homology on graphs, with publicly available Julia code and clear directions for further improvements.

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.

Efficient computations of discrete cubical homology

TL;DR

This work tackles the computational intractability of discrete cubical homology for graphs by combining mathematical and programming advances. The authors introduce a faster graph-map generation technique, leverage a quotient by the hyperoctahedral group to reduce module dimensions, and apply discrete-homotopy preprocessing to shrink graphs before computation. They also optimize matrix construction via degeneracy tracking, precomputed faces, and coordinate dictionaries, and demonstrate substantial speedups—enabling previously intractable cases such as high-dimensional homology and larger graphs, especially with parallelization. The result is the fastest known algorithm for discrete cubical homology on graphs, with publicly available Julia code and clear directions for further improvements.

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.

Paper Structure

This paper contains 30 sections, 8 theorems, 6 equations, 16 figures, 28 algorithms.

Table of Contents

  1. Discrete homology
  2. The naive algorithm
  3. Generation of maps
  4. Removal of degeneracies
  5. Matrix representation of the boundary map
  6. Putting it all together
  7. Analysis of computations
  8. Generation of singular n-cubes
  9. Pairing lower cubes to form higher ones
  10. Precomputing vertex neighborhoods
  11. Not storing the ($n+1$)-cubes
  12. Updated cubical homology algorithm
  13. Analysis of computations
  14. Degeneracies and generation of matrices
  15. Keeping track of degenerate coordinates
  16. ...and 15 more sections

Key Result

proposition 12

Let $G_0 = I_1^{\square n}$, with vertices relabelled as $(x_1, \dots, x_n, 0)$, and $G_1 = I_1^{\square n}$, with vertices relabelled as $(x_1, \dots, x_n, 1)$. Define the set and let $G$ to be the graph such that $G_V = {G_0}_V \sqcup {G_1}_V$, and $G_E = {G_0}_E \sqcup {G_1}_E \sqcup E$. Then $G = I_1^{\square n+1}$. ∎

Figures (16)

  • Figure 3: The graphs $I_1$, $I_1^{\square 2}$, and $I_1^{\square 3}$
  • Figure 4: Naive algorithm computation times
  • Figure 5: Naive algorithm setup times vs rank times
  • Figure 6: $I_1^{\square 2}$ paired with itself to form $I_1^{\square 3}$.
  • Figure 7: Updated algorithm computation times
  • ...and 11 more figures

Theorems & Definitions (34)

  • definition 1
  • example 2
  • definition 3
  • example 4
  • definition 5
  • definition 6
  • definition 7
  • definition 8
  • definition 9
  • definition 10
  • ...and 24 more