We present a dynamic data structure for maintaining the persistent homology of a time series of real numbers. The data structure supports local operations, including the insertion and deletion of an item and the cutting and concatenating of lists, each in time $O(\log n + k)$, in which $n$ counts the critical items and $k$ the changes in the augmented persistence diagram. To achieve this, we design a tailor-made tree structure with an unconventional representation, referred to as banana tree, which may be useful in its own right.

翻译：本文提出了一种动态数据结构，用于维护实数时间序列的持久同调。该数据结构支持局部操作，包括元素的插入与删除、列表的切割与拼接，每次操作时间复杂度为$O(\log n + k)$，其中$n$表示关键元素数量，$k$表示增广持久图的变化量。为实现这一目标，我们设计了一种具有非传统表示形式的定制树结构，称为香蕉树，该结构本身可能具有独立的应用价值。