Cofaces -- simplices that contain a given simplex -- have multiple important uses in generating and using a Vietoris-Rips filtration: both in creating the coboundary matrix for computing persistent cohomology, and for generating the ordered sequence of simplices in the first place. Traditionally, most methods have generated simplices first, and then sorted them in filtration order after the generation step. In this paper, we propose fast algorithms for generating the sequence of simplices by generating cofaces of a given simplex with the same diameter, which by construction produces simplices in filtration order, and for generating additional cofaces in filtration order using sorted neighborhood lists in order to generate coboundaries directly in filtration order.

翻译：共面——即包含给定单纯形的单纯形——在生成和使用 Vietoris-Rips 滤过的过程中具有多种重要用途：既可用于创建计算持久上同调的共边界矩阵，也可用于首先生成单纯形的有序序列。传统上，大多数方法首先生成单纯形，然后在生成步骤之后按滤过序对它们进行排序。在本文中，我们提出了快速算法，通过生成具有相同直径的给定单纯形的共面来生成单纯形序列，这种方法在构造上直接按滤过序产生单纯形；同时，我们还提出利用排序邻域列表按滤过序生成额外的共面，从而能够直接按滤过序生成共边界。