The Betti tables of a multigraded module encode the grades at which there is an algebraic change in the module. Multigraded modules show up in many areas of pure and applied mathematics, and in particular in topological data analysis, where they are known as persistence modules, and where their Betti tables describe the places at which the homology of filtered simplicial complexes changes. Although Betti tables of singly and bigraded modules are already being used in applications of topological data analysis, their computation in the bigraded case (which relies on an algorithm that is cubic in the size of the filtered simplicial complex) is a bottleneck when working with large datasets. We show that, in the special case of zero-dimensional homology (relevant for clustering and graph classification) Betti tables of bigraded modules can be computed in log-linear time. We also consider the problem of computing minimal presentations, and show that minimal presentations of zero-dimensional persistent homology can be computed in quadratic time, regardless of the grading poset.

翻译：多重分次模的Betti表编码了该模发生代数变化的次数。多重分次模出现在纯数学与应用数学的诸多领域，尤其在拓扑数据分析中被称为持续模，其Betti表描述了过滤单纯复形同调发生变化的位置。尽管单分次与双分次模的Betti表已在拓扑数据分析中得到应用，但双分次情形下的计算（依赖于一种在过滤单纯复形规模上呈立方复杂度的算法）在处理大规模数据集时成为瓶颈。我们证明，在零维同调（与聚类和图分类相关）这一特殊情形下，双分次模的Betti表可在对数线性时间内计算。同时我们考虑了极小表示的计算问题，并证明无论分次偏序集如何，零维持续同调的极小表示均可在平方时间内计算。