# Efficient algorithms for computing a minimal homology basis

arXiv: https://arxiv.org/abs/1801.06759 · HTML: https://mlsanity.com/paper/1801.06759/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jan 2018.

## Built on by (most cited first, 3 of 3)

- [Dist2Cycle: A Simplicial Neural Network for Homology Localization](https://mlsanity.com/paper/2110.15182/) (2021)
- [Minimal Cycle Representatives in Persistent Homology using Linear Programming: an Empirical Study with User's Guide](https://mlsanity.com/paper/2105.07025/) (2021)
- [Tight basis cycle representatives for persistent homology of large data sets](https://mlsanity.com/paper/2206.02925/) (2022)

## Abstract (arXiv)

Efficient computation of shortest cycles which form a homology basis under mathbbZ2-additions in a given simplicial complex K has been researched actively in recent years. When the complex K is a weighted graph with n vertices and m edges, the problem of computing a shortest (homology) cycle basis is known to be solvable in O(m2n/log n+ n2m)-time. Several works citeborradaile2017minimum, greedy have addressed the case when the complex K is a 2-manifold. The complexity of these algorithms depends on the rank g of the one-dimensional homology group of K. This rank g has a lower bound of Theta(n), where n denotes the number of simplices in K, giving an O(n4) worst-case time complexity for the algorithms in citeborradaile2017minimum,greedy. This worst-case complexity is improved in citeannotation to O(nω + n2gω-1) for general simplicial complexes where ω< 2.3728639 citele2014powers is the matrix multiplication exponent. Taking g=Theta(n), this provides an O(nω+1) worst-case algorithm. In this paper, we improve this time complexity. Combining the divide and conquer technique from citeDivideConquer with the use of annotations from citeannotation, we present an algorithm that runs in O(nω+n2g) time giving the first O(n3) worst-case algorithm for general complexes. If instead of minimal basis, we settle for an approximate basis, we can improve the running time even further. We show that a 2-approximate minimal homology basis can be computed in O(nω√n log n) expected time. We also study more general measures for defining the minimal basis and identify reasonable conditions on these measures that allow computing a minimal basis efficiently.