Fully Dynamic Strongly Connected Components in Planar Digraphs

TL;DR

The paper develops a sublinear fully dynamic data structure for maintaining SCCs in planar digraphs under planarity-preserving updates, achieving $ ilde{O}(n^{6/7})$ worst-case update time with an implicit SCC representation. The approach centers on dynamic $r$-divisions with few holes, reachability certificates, and a novel path net data structure that can list and aggregate vertices in an SCC-impacted region in near-linear time in the boundary size. It also provides a simpler substructure for dynamic strong connectivity with $ ilde{O}(n^{2/3})$ update time, and extends to a fully dynamic #SSR result with $ ilde{O}(n^{4/5})$ update time via additively weighted Voronoi diagrams. Together, these results show that planar digraphs permit substantially more efficient implicit SCC maintenance than general digraphs, and they offer practical mechanisms for reporting SCC contents and sizes, counting SCCs, and querying the SCC of a given vertex. The techniques advance the understanding of sublinear dynamic graph algorithms in sparse graph classes and suggest further exploration of embedding-respecting updates and related problems.

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.

Figures (1)

• Figure 1: Splitting the instance $(B,\Phi)$, where $B=S\cup T$, into 5 smaller instances with paths $\pi_1,\ldots,\pi_l$ (either originating or ending in $t_1$) for $l=4$. The vertices $S=\{s_1,\ldots,s_p\}$ are shown in blue, whereas the vertices $T=\{t_1,\ldots,t_q\}$ in red. The black arrows and dashed lines represent the individual parts of the curve $\Phi$: paths of the form $\pi_{u,v}$ or parts of the curve $h$, respectively. Note that the black arrows appear only on the $\Phi_{s_1,s_p}$ part of $\Phi$. The vertices $x_1,\ldots,x_4$, marked green, are precisely all the vertices of $S\setminus\{s_p\}$ that $t_1$ can reach or can be reached from. The obtained smaller instances are marked with distinct patterns. The instances marked with line patterns (types 1 or 2) are base instances. The instance marked using a dotted pattern (type 3) might constitute the only obtained instance that is not a base instance (for which the algorithm continues).

Theorems & Definitions (44)

• Theorem 1.1
• Lemma 1.1
• Theorem 2.1
• Lemma 3.1: Subramanian93
• Remark 3.2
• Lemma 3.2
• Lemma 3.2
• proof
• Lemma 3.2
• Definition 4.1