AI 中文总结
研究之字形代表更新问题，基于从\(R = DV\)分解提取之字形代表的算法，提出高效更新算法，克服邻接变化困难，能在二次时间内高效完成之字形代表更新。
AI 中文摘要
近年来，之字形持久性的计算取得了进展，结果表明许多问题的复杂度与非之字形情形紧密相关。目前主要的效率差距在于之字形代表的更新。本文基于一种从构造的非之字形的\(R = DV\)分解中提取之字形代表的近期算法，提出了高效的之字形代表更新算法。设计更新算法的主要困难在于在延长或缩短过滤的两个操作中发生的邻接变化。尽管存在邻接变化，但我们发现更新仍可在二次时间内高效完成。
英文摘要
Computation of zigzag persistence has progressed in recent years, with results showing that complexities of many problems closely align with those in the non-zigzag setting. The major efficiency gap now lies in the updating of zigzag representatives. In this paper, we propose efficient algorithms for updating zigzag representatives based on a recent algorithm for extracting zigzag representatives from a $R=DV$ decomposition of a constructed non-zigzag. The main difficulty for designing our update algorithms lies in the adjacency change occurring in two operations that elongate or shorten a filtration. Despite the adjacency change, we find that the update can still be done efficiently in quadratic time.
