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：藤蔓和葡萄园用于连接一组持续性图的堆栈，最初由Cohen-Steiner等人在非之字形设定中引入。我们考虑在之字形持久性的变化过滤中计算这些藤蔓，同时引入两种额外操作：扩张和收缩，以及非之字形设定中已有的置换操作。尽管在非之字形情况下，通过利用线性时间的置换操作可以在二次时间内实现扩张和收缩，但在之字形框架下如何以相同复杂度执行这些操作仍不明确。虽然利用最新的FastZigzag算法可以轻松在线性时间内实现置换操作，但扩张和收缩操作难以突破三次复杂度的障碍。我们的主要结果是：FastZigzag算法中半构造的上-下过滤结构确实可用于实现置换操作的线性时间复杂度和扩张/收缩操作的二次时间复杂度，从而匹配非之字形情况下所有对应操作的时间复杂度。