We give a fully dynamic algorithm maintaining a $(1-\varepsilon)$-approximate directed densest subgraph in $\tilde{O}(\log^3(n)/\varepsilon^6)$ amortized time or $\tilde{O}(\log^4(n)/\varepsilon^7)$ per edge update (where $\tilde{O}$ hides $\log\log$ factors), based on earlier work by Chekuri and Quanrud [arXiv:2210.02611]. This result improves on earlier work done by Sawlani and Wang [arXiv:1907.03037], which guarantees $O(\log^5(n)/\varepsilon^7)$ worst case time for edge insertions and deletions.

翻译：我们提出一种全动态算法，可在$\tilde{O}(\log^3(n)/\varepsilon^6)$均摊时间或每条边更新$\tilde{O}(\log^4(n)/\varepsilon^7)$时间内（其中$\tilde{O}$隐藏对数因子）维护$(1-\varepsilon)$-近似有向最密子图。该算法基于Chekuri与Quanrud的先前工作[arXiv:2210.02611]，相较于Sawlani与Wang[arXiv:1907.03037]的同类研究——其保证边插入与删除的最坏情况时间复杂度为$O(\log^5(n)/\varepsilon^7)$——实现了性能提升。