Vines and vineyard connecting a stack of persistence diagrams have been introduced in the non-zigzag setting by Cohen-Steiner et al. We consider computing these vines over changing filtrations for zigzag persistence while incorporating two more operations: expansions and contractions in addition to the transpositions considered in the non-zigzag setting. Although expansions and contractions can be implemented in quadratic time in the non-zigzag case by utilizing the linear-time transpositions, it is not obvious how they can be carried out under the zigzag framework with the same complexity. While transpositions alone can be easily conducted in linear time using the recent FastZigzag algorithm, expansions and contractions pose difficulty in breaking the barrier of cubic complexity. Our main result is that, the half-way constructed up-down filtration in the FastZigzag algorithm indeed can be used to achieve linear time complexity for transpositions and quadratic time complexity for expansions and contractions, matching the time complexity of all corresponding operations in the non-zigzag case.

翻译：摘要：Cohen-Steiner等人在非锯齿形情境下引入了连接持久性图堆栈的藤蔓与葡萄园结构。我们考虑在锯齿形持久性框架下，通过改变滤流计算此类藤蔓结构，并纳入两种额外操作：除非锯齿形情境中考虑的换位操作外，新增扩张与收缩操作。尽管在非锯齿形案例中，可利用线性时间换位操作以二次时间实现扩张与收缩，但在锯齿形框架下如何以相同复杂度执行这些操作尚不明确。仅换位操作可通过近期提出的FastZigzag算法在线性时间内轻松完成，但扩张与收缩却难以突破立方复杂度的壁垒。本文主要结论是：FastZigzag算法中半构建的上下滤流结构确实可用于实现换位操作的线性时间复杂度，以及扩张与收缩操作的二次时间复杂度，这与非锯齿形案例中所有对应操作的时间复杂度相匹配。