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Bootstrapping Dynamic APSP via Sparsification

Rasmus Kyng, Simon Meierhans, Gernot Zöcklein

TL;DR

This work tackles dynamic approximate All-Pairs Shortest Paths by bootstrapping a sparse, deterministic framework that mirrors Thorup–Zwick distance oracles while remaining fully dynamic. The key idea is to pick a small set of pivots and reduce distance computations to local neighborhoods and to distances among pivots, recursively maintaining these pivot distances on progressively smaller graphs. Core contributions include a dynamic KMG vertex sparsifier and a low-recourse dynamic spanner that can be bootstrapped together, enabling $m^{1+o(1)}$ total updates and $ ilde{O}(1)$ query time with $m^{o(1)}$-approximation; crucially, the approach avoids expander embeddings. The work also develops a robust theory of dynamic edge embeddings and patching to maintain spanners under updates, with batching schemes ensuring low sequential overhead. Together, these ideas yield a practical, fully-dynamic, deterministic APSP data structure with provable guarantees and scalable performance.

Abstract

We give a simple algorithm for the dynamic approximate All-Pairs Shortest Paths (APSP) problem. Given a graph $G = (V, E, l)$ with polynomially bounded edge lengths, our data structure processes $|E|$ edge insertions and deletions in total time $|E|^{1 + o(1)}$ and provides query access to $|E|^{o(1)}$-approximate distances in time $\tilde{O}(1)$ per query. We produce a data structure that mimics Thorup-Zwick distance oracles [TZ'05], but is dynamic and deterministic. Our algorithm selects a small number of pivot vertices. Then, for every other vertex, it reduces distance computation to maintaining distances to a small neighborhood around that vertex and to the nearest pivot. We maintain distances between pivots efficiently by representing them in a smaller graph and recursing. We construct these smaller graphs by (a) reducing vertex count using the dynamic distance-preserving core graphs of Kyng-Meierhans-Probst Gutenberg [KMPG'24] in a black-box manner and (b) reducing edge-count using a dynamic spanner akin to Chen-Kyng-Liu-Meierhans-Probst Gutenberg [CKL+'24]. Our dynamic spanner internally uses an APSP data structure. Choosing a large enough size reduction factor in the first step allows us to simultaneously bootstrap our spanner and a dynamic APSP data structure. Notably, our approach does not need expander graphs, an otherwise ubiquitous tool in derandomization.

Bootstrapping Dynamic APSP via Sparsification

TL;DR

This work tackles dynamic approximate All-Pairs Shortest Paths by bootstrapping a sparse, deterministic framework that mirrors Thorup–Zwick distance oracles while remaining fully dynamic. The key idea is to pick a small set of pivots and reduce distance computations to local neighborhoods and to distances among pivots, recursively maintaining these pivot distances on progressively smaller graphs. Core contributions include a dynamic KMG vertex sparsifier and a low-recourse dynamic spanner that can be bootstrapped together, enabling total updates and query time with -approximation; crucially, the approach avoids expander embeddings. The work also develops a robust theory of dynamic edge embeddings and patching to maintain spanners under updates, with batching schemes ensuring low sequential overhead. Together, these ideas yield a practical, fully-dynamic, deterministic APSP data structure with provable guarantees and scalable performance.

Abstract

We give a simple algorithm for the dynamic approximate All-Pairs Shortest Paths (APSP) problem. Given a graph with polynomially bounded edge lengths, our data structure processes edge insertions and deletions in total time and provides query access to -approximate distances in time per query. We produce a data structure that mimics Thorup-Zwick distance oracles [TZ'05], but is dynamic and deterministic. Our algorithm selects a small number of pivot vertices. Then, for every other vertex, it reduces distance computation to maintaining distances to a small neighborhood around that vertex and to the nearest pivot. We maintain distances between pivots efficiently by representing them in a smaller graph and recursing. We construct these smaller graphs by (a) reducing vertex count using the dynamic distance-preserving core graphs of Kyng-Meierhans-Probst Gutenberg [KMPG'24] in a black-box manner and (b) reducing edge-count using a dynamic spanner akin to Chen-Kyng-Liu-Meierhans-Probst Gutenberg [CKL+'24]. Our dynamic spanner internally uses an APSP data structure. Choosing a large enough size reduction factor in the first step allows us to simultaneously bootstrap our spanner and a dynamic APSP data structure. Notably, our approach does not need expander graphs, an otherwise ubiquitous tool in derandomization.
Paper Structure (37 sections, 14 theorems, 10 equations, 1 figure, 6 algorithms)

This paper contains 37 sections, 14 theorems, 10 equations, 1 figure, 6 algorithms.

Table of Contents

  1. Introduction
  2. Roadmap.
  3. Preliminaries
  4. Graph Notation.
  5. Paths, Distances and Balls.
  6. Dynamic Graphs.
  7. Dynamic Objects and Recourse.
  8. Dynamic APSP.
  9. Degree Reduction Through Binary Search Trees
  10. Warm up: Low-Recourse Thorup-Zwick'05
  11. Pivot Hierarchies.
  12. Low-Recourse KMG-vertex sparsifier.
  13. Low-Recourse Dynamic Spanner.
  14. Low-Recourse Pivot Hierarchies via Iterated Sparsification.
  15. The Algorithm
  16. ...and 22 more sections

Key Result

Theorem 1.1

For a graph $G = (V, E, \boldsymbol{\mathit{l}})$ with polynomially bounded integral edge lengths $\boldsymbol{\mathit{l}} \in \mathbb{R}^E_{> 0}$ undergoing up to $m$ edge updates (insertions and deletions), there is an algorithm with total update time $m^{1 + o(1)}$ that maintains query access to The query time is $\widetilde{O}(1)$.

Figures (1)

  • Figure 1: The black graph $H$ is a spanner of the graph $G$ that also contains the red and blue edge. After the central edge is removed, the red and blue edge are projected onto the graph $J'$ supported on the two vertices incident to the deletion. Then this graph is sparsified, and the spanner $H$ is patched with the remaining blue edge. The embedding path of the red edge is repaired using the previous blue path and the new edge. When sparsifying $J'$ we ensure that the extra vertex congestion caused by these repaired paths in $H$ only increases by a small sub-polynomial factor.

Theorems & Definitions (57)

  • Theorem 1.1
  • Definition 2.1: Spanner
  • Definition 2.2: Vertex Split
  • Definition 2.3: Master Nodes
  • Definition 2.4: fully-dynamic Graph
  • Definition 2.5: Edge-Dynamic Graph
  • Definition 2.6: Dynamic APSP
  • Definition 3.1: Pivot Hierarchy
  • Definition 3.2: Distances
  • Theorem 3.3
  • ...and 47 more