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

2026-06-01

AI Summary

This work addresses the offline all-pairs shortest paths (APSP) problem in planar directed graphs under dynamic edge-weight updates. The authors propose an efficient maintenance algorithm based on dense distance graphs (DDGs), leveraging the structural properties of planar graphs and a divide-and-conquer strategy. Their approach achieves, for the first time in an offline setting, a dynamic maintenance complexity strictly better than recomputation from scratch, and reduces general dynamic APSP to the decremental case. The designed data structure attains Õ(√n) worst-case update and query time—up to polylogarithmic factors—approaching known theoretical lower bounds and significantly advancing the state of the art in dynamic graph algorithms.

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 $x,y \in V(G)$. The currently best algorithms for this problem require $\tilde{O}(n^{2/3})$ worst-case update and query time, while conditional lower bounds show that either update or query time $n^{0.5-δ}$ is needed for any constant $δ> 0$. In this article, we present the first algorithm with near-optimal $\tilde{O}(\sqrt{n})$ 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 \emph{online} algorithm for the incremental APSP problem with $\tilde{O}(\sqrt{n})$ 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.

Problem

• dynamic all-pairs shortest paths
• planar digraphs
• edge weight updates
• distance queries

Innovation

• dynamic APSP
• planar digraphs
• dense distance graphs
• offline algorithm
• near-optimal complexity