This paper introduces an efficient algorithm for persistence diagram computation, given an input piecewise linear scalar field $f$ defined on a $d$-dimensional simplicial complex $K$, with $d \leq 3$. Our work revisits the seminal algorithm "PairSimplices" [31], [103] with discrete Morse theory (DMT) [34], [80], which greatly reduces the number of input simplices to consider. Further, we also extend to DMT and accelerate the stratification strategy described in "PairSimplices" for the fast computation of the $0^{th}$ and $(d - 1)^{th}$ diagrams, noted $D_0(f)$ and $D_{d-1}(f)$. Minima-saddle persistence pairs ($D_0(f)$) and saddle-maximum persistence pairs ($D_{d-1}(f)$) are efficiently computed by processing, with a Union-Find, the unstable sets of $1$-saddles and the stable sets of $(d - 1)$-saddles. This fast pre-computation for the dimensions $0$ and $(d - 1)$ enables an aggressive specialization of [4] to the 3D case, which results in a drastic reduction of the number of input simplices for the computation of $D_1(f)$, the intermediate layer of the sandwich. Finally, we document several performance improvements via shared-memory parallelism. We provide an open-source implementation of our algorithm for reproducibility purposes. We also contribute a reproducible benchmark package, which exploits three-dimensional data from a public repository and compares our algorithm to a variety of publicly available implementations. Extensive experiments indicate that our algorithm improves by two orders of magnitude the time performance of the seminal "PairSimplices" algorithm it extends. Moreover, it also improves memory footprint and time performance over a selection of 14 competing approaches, with a substantial gain over the fastest available approaches, while producing a strictly identical output.

翻译：本文引入了一种高效的耐久性图计算算法, 此外, 我们还向 DMT 扩展并加速“ 直线缩放” 中描述的分层战略, 用于快速计算 $0 =th] 美元和 $(d - 1) 美元 3美元。 我们的工作重新审视了“ PairSimplices” 的原始算法 [31], [103] 与离散的摩斯理论(DMT) [34], [80], 大大降低了要考虑的进缩缩图计算数量。 此外, 我们还扩展到 DMTMTMT, 加速了“PairSimplical ” 中描述的分层战略, 快速计算了 $(d_th) 美元 和 $(d - 1) 美元 美元 的中间平流利值 图表。 微量的计算结果也通过处理, 快速更新了我们 3美元 的递增的运行过程。