A Near-Optimal Offline Algorithm for Dynamic All-Pairs Shortest Paths in Planar Digraphs

Abstract

In the planar, dynamic All-Pairs Shortest Paths (APSP) problem, a planar, weighted digraph G undergoes a sequence of edge weight updates and the goal is to maintain a data structure on G, that can quickly answer distance queries between any two vertices x,y ∊ V(G). The currently best algorithms [FOCS'01, SODA'05] for this problem require Õ(n2/3) worst-case update and query time, while conditional lower bounds [FOCS'16] show that either update or query time is needed1. In this article, we present the first algorithm with near-optimal worst-case update and query time for the offline setting, where the update sequence is given initially. This result is obtained by giving the first offline dynamic algorithm for maintaining dense distance graphs (DDGs) faster than recomputing from scratch after each update. Further, we also present an online algorithm for the incremental APSP problem with worst-case update/query time. This allows us to reduce the online dynamic APSP problem to the online decremental APSP problem, which constitutes partial progress even for the online version of this notorious problem.