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Sparse Bounded Hop-Spanners for Geometric Intersection Graphs

Authors Sujoy Bhore , Timothy M. Chan , Zhengcheng Huang , Shakhar Smorodinsky , Csaba D. Tóth

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  • Theory of computation → Computational geometry
Keywords
  • Geometric Spanners
  • Geometric Intersection Graphs

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Abstract

We present new results on 2- and 3-hop spanners for geometric intersection graphs. These include improved upper and lower bounds for 2- and 3-hop spanners for many geometric intersection graphs in ℝ^d. For example, we show that the intersection graph of n balls in ℝ^d admits a 2-hop spanner of size O^*(n^{3/2 - 1/(2(2⌊d/2⌋ + 1))}) and the intersection graph of n fat axis-parallel boxes in ℝ^d admits a 2-hop spanner of size O(n log^{d+1}n). 
Furthermore, we show that the intersection graph of general semi-algebraic objects in ℝ^d admits a 3-hop spanner of size O^*(n^{3/2 - 1/(2(2D-1))}), where D is a parameter associated with the description complexity of the objects. For such families (or more specifically, for tetrahedra in ℝ³), we provide a lower bound of Ω(n^{4/3}). For 3-hop and axis-parallel boxes in ℝ^d, we provide the upper bound O(n log ^{d-1}n) and lower bound Ω(n ({log n}/{log log n})^{d-2}).

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Sujoy Bhore, Timothy M. Chan, Zhengcheng Huang, Shakhar Smorodinsky, and Csaba D. Tóth. Sparse Bounded Hop-Spanners for Geometric Intersection Graphs. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 17:1-17:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.17

Author Details

Sujoy Bhore
  • Department of Computer Science and Engineering, Indian Institute of Technology Bombay, India
Timothy M. Chan
  • Siebel School of Computing and Data Science, University of Illinois at Urbana-Champaign, IL, USA
Zhengcheng Huang
  • Siebel School of Computing and Data Science, University of Illinois at Urbana-Champaign, IL, USA
Shakhar Smorodinsky
  • Department of Computer Science, Ben-Gurion University of the Negev, Be'er Sheva, Israel
Csaba D. Tóth
  • Department of Mathematics, California State University Northridge, Los Angeles, CA, USA
  • Department of Computer Science, Tufts University, Medford, MA, USA

Funding

  • Chan, Timothy M.: Work supported in part by NSF Grant CCF-2224271.
  • Smorodinsky, Shakhar: Partially supported by Grant 1065/20 from the Israel Science Foundation, by the United States – Israel Binational Science Foundation (NSF-BSF grant no. 2022792) and by ERC grant no. 882971 "GeoScape" and by the Erdős Center.
  • Tóth, Csaba D.: Research was partially supported by the NSF award DMS-2154347.

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