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Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity

Authors Sujoy Bhore , Sándor Kisfaludi‑Bak , Lazar Milenković , Csaba D. Tóth , Karol Węgrzycki , Sampson Wong

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Subject Classification

ACM Subject Classification
  • Theory of computation → Sparsification and spanners
  • Theory of computation → Routing and network design problems
  • Mathematics of computing → Graph algorithms
Keywords
  • geometric network design
  • spanners
  • crossing number
  • incidences

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Abstract

A Euclidean noncrossing Steiner (1+ε)-spanner for a point set P ⊂ ℝ² is a planar straight-line graph that, for any two points a, b ∈ P, contains a path whose length is at most 1+ε times the Euclidean distance between a and b. We construct a Euclidean noncrossing Steiner (1+ε)-spanner with O(n/ε^{3/2}) edges for any set of n points in the plane. This result improves upon the previous best upper bound of O(n/ε⁴) obtained nearly three decades ago. We also establish an almost matching lower bound: There exist n points in the plane for which any Euclidean noncrossing Steiner (1+ε)-spanner has Ω_μ(n/ε^{3/2-μ}) edges for any μ > 0. Our lower bound uses recent generalizations of the Szemerédi-Trotter theorem to disk-tube incidences in geometric measure theory.

Cite As Get BibTex

Sujoy Bhore, Sándor Kisfaludi‑Bak, Lazar Milenković, Csaba D. Tóth, Karol Węgrzycki, and Sampson Wong. Euclidean Noncrossing Steiner Spanners of Nearly Optimal Sparsity. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 15:1-15:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.15

Author Details

Sujoy Bhore
  • Department of Computer Science & Engineering, Indian Institute of Technology Bombay, Mumbai, India
Sándor Kisfaludi‑Bak
  • Aalto University, Espoo, Finland
Lazar Milenković
  • Tel Aviv University, Israel
  • INSAIT, Sofia University "St. Kliment Ohridski", Bulgaria
Csaba D. Tóth
  • Department of Mathematics, California State University Northridge, Los Angeles, CA, USA
  • Department of Computer Science, Tufts University, Medford, MA, USA
Karol Węgrzycki
  • Max Planck Institute for Informatics, Saarbrücken, Germany
Sampson Wong
  • University of Copenhagen, Denmark

Funding

  • Bhore, Sujoy: Work supported in part by ANRF ARG-MATRICS, Grant 002465.
  • Kisfaludi‑Bak, Sándor: Supported by the Research Council of Finland, Grant 363444.
  • Milenković, Lazar: Funded by a grant from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel, and the United States National Science Foundation (NSF). This research was partially funded by the Ministry of Education and Science of Bulgaria (support for INSAIT, part of the Bulgarian National Roadmap for Research Infrastructure).
  • Tóth, Csaba D.: Research supported, in part, by the NSF award DMS-2154347.
  • Węgrzycki, Karol: Supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) grant number 559177164.
  • Wong, Sampson: Supported by the European Union’s Marie Skłodowska-Curie Actions Postdoctoral Fellowship, grant number 101146276.

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