Efficient algorithms for computing a minimal homology basis

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 andm edges, the problem of computing a shortest (homology) cycle basis is known to be solvable in O(mn/ log n+nm)-time. Several works [1,2] 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(n) worst-case time complexity for the algorithms in [1,2]. This worst-case complexity is improved in [3] to O(n+ng) for general simplicial complexes where ω < 2.3728639 [4] is the matrix multiplication exponent. Taking g = Θ(n), this provides an O(n) worst-case algorithm. In this paper, we improve this time complexity. Combining the divide and conquer technique from [5] with the use of annotations from [3], we present an algorithm that runs in O(n + ng) time giving the first O(n) 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 √ 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.