0-dimensional persistent homology is known, from a computational point of view, as the easy case. Indeed, given a list of $n$ edges in non-decreasing order of filtration value, one only needs a union-find data structure to keep track of the connected components and we get the persistence diagram in time $O(n\alpha(n))$. The running time is thus usually dominated by sorting the edges in $\Theta(n\log(n))$. A little-known fact is that, in the particularly simple case of studying the sublevel sets of a piecewise-linear function on $\mathbb{R}$ or $\mathbb{S}^1$, persistence can actually be computed in linear time. This note presents a simple algorithm that achieves this complexity and an extension to image persistence. An implementation is available in Gudhi.

翻译：从计算角度来看，零维持续同调被认为是最简单的情况。实际上，给定一个按滤值非递减顺序排列的$n$条边列表，我们只需要一个并查集数据结构来跟踪连通分量，即可在$O(n\alpha(n))$时间内获得持续图。因此，运行时间通常由$\Theta(n\log(n))$的边排序步骤主导。一个鲜为人知的事实是，在研究$\mathbb{R}$或$\mathbb{S}^1$上的分段线性函数子水平集这一特别简单的情形时，实际上可以在线性时间内计算持续同调。本文提出了一种实现该复杂度特性的简单算法，并将其扩展至图像持续同调。该算法已在Gudhi库中实现。