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A Simple Dynamic Spanner via APSP

Rasmus Kyng, Simon Meierhans, Gernot Zöcklein

TL;DR

The paper tackles dynamic maintenance of sparse graph spanners under edge updates by reducing the problem to dynamic All-Pairs Shortest Paths (APSP). It introduces a simple greedy-spanner framework and two variants: a low-recourse version and a more efficient one that leverages a path-reporting APSP oracle alongside edge embeddings to control reinsertion work. The main results show a $2\gamma\log n$-approximate spanner maintained with total update time $O(\beta\gamma m\log n + \alpha\gamma n\log n)$ and total recourse $O(\gamma\alpha n\log n)$, where the factors $\gamma,\alpha,\beta$ are subpolynomial in $n$. This approach implies sublinear recourse when insertions dominate deletions and indicates that improvements in APSP directly translate to spanner improvements.

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.

A Simple Dynamic Spanner via APSP

TL;DR

The paper tackles dynamic maintenance of sparse graph spanners under edge updates by reducing the problem to dynamic All-Pairs Shortest Paths (APSP). It introduces a simple greedy-spanner framework and two variants: a low-recourse version and a more efficient one that leverages a path-reporting APSP oracle alongside edge embeddings to control reinsertion work. The main results show a -approximate spanner maintained with total update time and total recourse , where the factors are subpolynomial in . This approach implies sublinear recourse when insertions dominate deletions and indicates that improvements in APSP directly translate to spanner improvements.

Abstract

We give a simple algorithm for maintaining a -approximate spanner of a graph with vertices as receives edge updates by reduction to the dynamic All-Pairs Shortest Paths (APSP) problem. Given an initially empty graph , our algorithm processes insertions and deletions in total time and maintains an initially empty spanner with total recourse . 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 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.
Paper Structure (10 sections, 6 theorems, 1 algorithm)

This paper contains 10 sections, 6 theorems, 1 algorithm.

Table of Contents

  1. Introduction
  2. Prior Work.
  3. Preliminaries
  4. Static and Dynamic Graphs.
  5. Edge Embeddings.
  6. Dynamic All-Pairs Shortest Paths
  7. A Dynamic Greedy Spanner
  8. A Simple Low-Recourse Algorithm.
  9. A Simple Efficient Algorithm.
  10. Analysis.

Key Result

Theorem 1.2

Given an initially empty edge-dynamic graph $G = (V, E)$ on $n = |V|$ vertices receiving $m$ edge insertions and up to $n$ edge deletions where $m \geq n$, there is an algorithm maintaining a $n^{o(1)}$-approximate spanner $H$ with total update time $m^{1 + o(1)}$ and total recourse $n^{1 + o(1)}$.

Theorems & Definitions (17)

  • proof
  • Theorem 1.2
  • Definition 3.1: APSP
  • Lemma 4.1
  • proof
  • Claim 4.2
  • proof
  • Corollary 4.3
  • proof
  • Claim 4.4
  • ...and 7 more