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：Cohen-Steiner等人已在非曲折情形下引入了连接持久性图堆的藤蔓与葡萄园。本文考虑在曲折持久性中，针对变化过滤计算这些藤蔓，并额外纳入两种操作：扩张与收缩，以及非曲折情形下已有的转置操作。尽管在非曲折情形下，利用线性时间转置操作可实现二次时间复杂度的扩张与收缩，但在曲折框架下如何以相同复杂度实现这些操作并不明确。虽然单独的转置操作可通过近期提出的FastZigzag算法在线性时间内轻松完成，但扩张与收缩难以突破立方复杂度的障碍。我们的主要结果是：FastZigzag算法中半构造的上下过滤结构，确实可用于实现转置的线性时间复杂度和扩张与收缩的二次时间复杂度，从而匹配非曲折情形下所有对应操作的时间复杂度。