In this paper, we introduce a fast and memory efficient approach to compute the Persistent Homology (PH) of a sequence of simplicial complexes. The basic idea is to simplify the complexes of the input sequence by using strong collapses, as introduced by Barmak and Miniam [DCG (2012)], and to compute the PH of an induced sequence of reduced simplicial complexes that has the same PH as the initial one. Our approach has several salient features that distinguishes it from previous work. It is not limited to filtrations (i.e. sequences of nested simplicial subcomplexes) but works for other types of sequences like towers and zigzags. To strong collapse a simplicial complex, we only need to store the maximal simplices of the complex, not the full set of all its simplices, which saves a lot of space and time. Moreover, the complexes in the sequence can be strong collapsed independently and in parallel. We also focus on the problem of computing persistent homology of a flag tower, i.e. a sequence of flag complexes connected by simplicial maps. We show that if we restrict the class of simplicial complexes to flag complexes, we can achieve decisive improvement in terms of time and space complexities with respect to previous work. Moreover we can strong collapse a flag complex knowing only its 1-skeleton and the resulting complex is also a flag complex. When we strong collapse the complexes in a flag tower, we obtain a reduced sequence that is also a flag tower we call the core flag tower. We then convert the core flag tower to an equivalent filtration to compute its PH. Here again, we only use the 1-skeletons of the complexes. The resulting method is simple and extremely efficient. As a result and as demonstrated by numerous experiments on publicly available data sets, our approach is extremely fast and memory efficient in practice. Finally, we can compromise between precision and time by choosing the number of simplicial complexes of the sequence we strong collapse.

中文翻译：

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强塌陷和持久同源

在本文中，我们介绍了一种快速且内存高效的方法来计算一系列单纯复形的持久同源性 (PH)。其基本思想是通过使用强塌陷来简化输入序列的复形，正如 Barmak 和 Miniam [DCG (2012)] 所介绍的，并计算具有相同 PH 的简化单纯复形的诱导序列的 PH最初的一个。我们的方法有几个显着的特点，将其与以前的工作区分开来。它不限于过滤（即嵌套单纯子复形的序列），还适用于其他类型的序列，如塔和之字形。要对单纯复形进行强坍缩，我们只需要存储复形的最大单纯形，而不是其所有单纯形的完整集合，这样可以节省大量空间和时间。而且，序列中的配合物可以独立和平行地强烈塌缩。我们还关注计算标志塔的持久同源性问题，即通过单纯映射连接的标志复合体序列。我们表明，如果我们将单纯复形的类别限制为标记复形，我们可以在时间和空间复杂性方面取得相对于先前工作的决定性改进。此外，我们可以强折叠一个只知道它的 1 骨架的标志复合体，并且生成的复合体也是一个标志复合体。当我们强折叠旗塔中的复合体时，我们得到一个简化的序列，它也是我们称之为核心旗塔的旗塔。然后我们将核心标志塔转换为等效过滤来计算其 PH。再次，我们只使用复合体的 1 骨架。所得方法简单且极其有效。因此，正如对公开可用数据集的大量实验所证明的那样，我们的方法在实践中非常快速且内存高效。最后，我们可以通过选择强坍缩序列的单纯复形的数量来在精度和时间之间进行折衷。