A Simple Dynamic Spanner via APSP

TL;DR

This paper presents a simple, efficient algorithm for maintaining an approximate graph spanner dynamically, leveraging reductions to the APSP problem, with performance closely tied to improvements in APSP algorithms.

Contribution

It introduces a straightforward reduction-based method for dynamic spanner maintenance with near-optimal runtime and recourse, connecting improvements in APSP to better spanners.

Findings

Achieves $n^{o(1)}$-approximate spanner with efficient update times.

Maintains spanner with total recourse $n^{1+o(1)}$ over updates.

Runtime overhead is only a logarithmic factor over APSP data structures.

Abstract

We give a simple algorithm for maintaining a $n^{o(1)}$-approximate spanner $H$ of a graph $G$ with $n$ vertices as $G$ receives edge updates by reduction to the dynamic All-Pairs Shortest Paths (APSP) problem. Given an initially empty graph $G$, our algorithm processes $m$ insertions and $n$ deletions in total time $m^{1 + o(1)}$ and maintains an initially empty spanner $H$ with total recourse $n^{1 + o(1)}$. When the number of insertions is much larger than the number of deletions, this notably yields recourse sub-linear in the total number of updates. Our algorithm only has a single $O(\log n)$ factor overhead in runtime and approximation compared to the underlying APSP data structure. Therefore, future improvements for APSP will directly yield an improved dynamic spanner.