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A Dynamic (1+ε)-Spanner for Disk Intersection Graphs

Authors Sarita de Berg , Ivor van der Hoog , Eva Rotenberg , Johanne Müller Vistisen , Sampson Wong

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • intersection graphs
  • dynamic data structures
  • spanners

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Abstract

We maintain a (1+ε)-spanner over the disk intersection graph of a dynamic set of disks. We restrict all disks to have their diameter in [4,Ψ] for some fixed and known Ψ. The resulting (1+ε)-spanner has size O(n ε^{-2} log Ψ log(ε^{-1})), where n is the present number of disks.
We develop a novel use of persistent data structures to dynamically maintain our (1+ε)-spanner. Our approach requires O(ε^{-2} n log⁴n log Ψ) space and has an O((Ψ/ε)² log⁴n log²Ψ log²(ε^{-1})) expected amortised update time. For constant ε and Ψ, this spanner has near-linear size, uses near-linear space and has polylogarithmic update time. Furthermore, we observe that for any ε < 1, our spanner also serves as a connectivity data structure. With a slight adaptation of our techniques, this leads to better bounds for dynamically supporting connectivity queries in a disk intersection graph. In particular, we improve the space usage when compared to the dynamic data structure of (Baumann et al., DCG'24), replacing the linear dependency on Ψ by a polylogarithmic dependency. Finally, we generalise our results to d-dimensional hypercubes.

Cite As Get BibTex

Sarita de Berg, Ivor van der Hoog, Eva Rotenberg, Johanne Müller Vistisen, and Sampson Wong. A Dynamic (1+ε)-Spanner for Disk Intersection Graphs. In 34th Annual European Symposium on Algorithms (ESA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 388, pp. 52:1-52:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ESA.2026.52

Author Details

Sarita de Berg
  • IT University of Copenhagen, Denmark
Ivor van der Hoog
  • IT University of Copenhagen, Denmark
Eva Rotenberg
  • IT University of Copenhagen, Denmark
Johanne Müller Vistisen
  • IT University of Copenhagen, Denmark
Sampson Wong
  • University of Copenhagen, Denmark

Funding

This research was supported by Danmarks Frie Forskningsfond Case no. 10.46540/3160-00020B, the VILLUM Foundation grant (VIL37507) "Efficient Recomputations for Changeful Problems", and the European Union’s Marie Skłodowska-Curie Actions Postdoctoral Fellowship Project number No. 101146276.

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