Efficient algorithms for computing a minimal homology basis

2018/01/21 by Tamal K. Dey, Dey, Tamal K., Tianqi Li +3
Computer Science · Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #Algebraic Topology (math.AT) #Combinatorics (math.CO) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.1801.06759

openalex publication_date 2018/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Efficient computation of shortest cycles which form a homology basis under ℤ₂-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 Θ(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(n^ω+ n²g^ω-1) for general simplicial complexes where ω< 2.3728639 \citele2014powers is the matrix multiplication exponent. Taking g=Θ(n), this provides an O(n^ω+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(n^ω+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(n^ω√(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.

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