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

We study the fully-dynamic all-pair shortest paths (APSP) problem on planar graphs: given an n-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+ϵ)-approximate} dynamic algorithm that supports edge updates and distance queries in n^o(1) time, for any 1/poly(log n) < ϵ < 1. Our result is a significant improvement over the best previously known bound of Õ(√{n}) on update and query time due to [Abraham, Chechik, and Gavoille, STOC '12], and bypasses a Ω(√{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].