In this paper, we consider maintaining strongly connected components (SCCs) of a directed planar graph subject to edge insertions and deletions. We show a data structure maintaining an implicit representation of the SCCs within $\tilde{O}(n^{6/7})$ worst-case time per update. The data structure supports, in $O(\log^2{n})$ time, reporting vertices of any specified SCC (with constant overhead per reported vertex) and aggregating vertex information (e.g., computing the maximum label) over all the vertices of that SCC. Furthermore, it can maintain global information about the structure of SCCs, such as the number of SCCs or the size of the largest SCC. To the best of our knowledge, no fully dynamic SCCs data structures with sublinear update time have been previously known for any major subclass of digraphs. Our result should be contrasted with the known $n^{1-o(1)}$ amortized update time lower bound conditional on SETH, which holds even for dynamically maintaining whether a general digraph has more than two SCCs.

翻译：本文研究在有向平面图中维护强连通分量（SCCs）的问题，该图允许边插入与删除操作。我们提出了一种数据结构，能在每次更新的最坏情况 $\tilde{O}(n^{6/7})$ 时间内维持 SCCs 的一种隐式表示。该数据结构支持在 $O(\log^2{n})$ 时间内报告任意指定 SCC 的顶点（每个被报告顶点的额外开销为常数）以及在该 SCC 的所有顶点上聚合顶点信息（例如计算最大标签）。此外，它还能维护关于 SCCs 结构的全局信息，例如 SCC 的数量或最大 SCC 的大小。据我们所知，此前在有向图的任何主要子类中，均未出现具有亚线性更新时间的完全动态 SCCs 数据结构。我们的结果应与已知的在 SETH 条件制约下的 $n^{1-o(1)}$ 摊销更新时间下界形成对比，该下界甚至适用于动态判断一般有向图是否具有多于两个 SCC 的问题。