Near-Optimal (1+ε)-Approximate Fully-Dynamic All-Pairs Shortest Paths in Planar Graphs

Arnold Filtser, Gramoz Goranci, Neel Patel, Maximilian Probst Gutenberg

Abstract: We study the fully-dynamic all-pair shortest paths (APSP) problem on planar graphs: given an $n-\mathbf{vertex}$ planar graph $G=(V, E)$ undergoing edge insertions and deletions, the goal is to efficiently process these updates and support distance and shortest path queries. We give a $(1+\epsilon)-\mathbf{approximate}$ dynamic algorithm that supports edge updates and distance queries in $n^{o(1)}$ time, for any $1/\mathbf{poly}(\log n) < \epsilon < 1$ . Our result is a significant improvement over the best previously known bound of $\tilde{O}(\sqrt{n})$ on update and query time due to [Abraham, Chechik, and Gavoille, STOC ’12], and bypasses a $\Omega(\sqrt{n})$ conditional lower-bound on update and query time for exact fully dynamic planar APSP [Abboud and Dahlgaard, FOCS ’16]. The main technical contribution behind our result is to dynamize the planar emulator construction due to [Chang, Krauthgamer, Tan, STOC ’22].