Matrix reduction is the standard procedure for computing the persistent homology of a filtered simplicial complex with $m$ simplices. Its output is a particular decomposition of the total boundary matrix, from which the persistence diagrams and generating cycles are derived. Persistence diagrams are known to vary continuously with respect to their input, motivating the study of their computation for time-varying filtered complexes. Computing persistence dynamically can be reduced to maintaining a valid decomposition under adjacent transpositions in the filtration order. Since there are $O(m^2)$ such transpositions, this maintenance procedure exhibits limited scalability and is often too fine for many applications. We propose a coarser strategy for maintaining the decomposition over a 1-parameter family of filtrations. By reduction to a particular longest common subsequence problem, we show that the minimal number of decomposition updates $d$ can be found in $O(m \log \log m)$ time and $O(m)$ space, and that the corresponding sequence of permutations -- which we call a schedule -- can be constructed in $O(d m \log m)$ time. We also show that, in expectation, the storage needed to employ this strategy is actually sublinear in $m$. Exploiting this connection, we show experimentally that the decrease in operations to compute diagrams across a family of filtrations is proportional to the difference between the expected quadratic number of states and the proposed sublinear coarsening. Applications to video data, dynamic metric space data, and multiparameter persistence are also presented.

翻译：矩阵约简是计算具有$m$个单形过滤单纯复形的持续同调的标准流程。其输出是总边界矩阵的一种特定分解，由此可导出持续同调图和生成循环。已知持续同调图相对于输入连续变化，这推动了对其随时间变化过滤复形计算的研究。动态计算持续同调可归结为在过滤顺序的相邻转置下维护有效分解。由于存在$O(m^2)$个此类转置，这种维护过程扩展性有限，且对许多应用而言过于精细。我们提出一种粗粒度策略，用于在单参数过滤族上维护分解。通过将其归结为特定最长公共子序列问题，我们证明最小分解更新次数$d$可在$O(m \log \log m)$时间和$O(m)$空间内找到，且相应的置换序列——我们称之为调度——可在$O(d m \log m)$时间内构建。我们还证明，在期望情况下，采用该策略所需存储量实际上是$m$的次线性。利用此关联，我们通过实验表明，在过滤族上计算同调图时操作次数的减少正比于预期二次状态数与所提出次线性粗粒度之间的差异。此外，还展示了该方法在视频数据、动态度量空间数据以及多参数持续同调中的应用。