Fully Dynamic Strongly Connected Components in Planar Digraphs

TL;DR

This paper introduces a novel data structure for maintaining strongly connected components in dynamic planar graphs with sublinear worst-case update time, enabling efficient updates and queries on SCCs.

Contribution

It presents the first fully dynamic SCC data structure for planar digraphs with sublinear worst-case update time, improving over previous bounds.

Findings

Supports implicit SCC representation with $ ilde{O}(n^{6/7})$ update time.

Enables fast reporting and aggregation of SCC vertex information.

Maintains global SCC structure data efficiently.

Abstract

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.