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Charting the Diameter Computation Landscape of Intersection Graphs in 3D and Above

Authors Timothy M. Chan , Hsien-Chih Chang , Jie Gao , Sándor Kisfaludi-Bak , Hung Le , Da Wei Zheng

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LIPIcs.SoCG.2026.29.pdf
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ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Graph Diameter
  • Geometric Intersection Graphs
  • Unit Ball Graphs

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Abstract

Recent research on computing the diameter of geometric intersection graphs has made significant strides, primarily focusing on the 2D case [Duraj et al., 2024; Hsien-Chih Chang et al., 2024; Chan et al., 2025] where truly subquadratic-time algorithms were given for simple objects such as unit-disks and (axis-aligned) squares. However, in three or higher dimensions, there is no known truly subquadratic-time algorithm for any intersection graph of non-trivial objects, even basic ones such as unit balls or (axis-aligned) unit cubes. This was partially explained by the pioneering work of Bringmann et al. [Karl Bringmann et al., 2022] which gave several truly subquadratic lower bounds, notably for unit balls or unit cubes in 3D when the graph diameter Δ is at least Ω(log n), hinting at a pessimistic outlook for the complexity of the diameter problem in higher dimensions. In this paper, we substantially extend the landscape of diameter computation for objects in three and higher dimensions, giving a few positive results. Our highlighted findings include:  
1) A truly subquadratic-time algorithm for deciding if the diameter of unit cubes in 3D is at most 3 (Diameter-3 hereafter), the first algorithm of its kind for objects in 3D or higher dimensions. Our algorithm is based on a novel connection to pseudolines, which is of independent interest. 
2) A truly subquadratic time lower bound for Diameter-3 of unit balls in 3D under the Orthogonal Vector (OV) hypothesis, giving the first separation between unit balls and unit cubes in the small diameter regime. Previously, computing the diameter for both objects was known to be quadratic hard when the diameter is Ω(log n) [Karl Bringmann et al., 2022]. 
3) A near-linear-time algorithm for Diameter-2 of unit cubes in 3D, generalizing the previous result for unit squares in 2D [Karl Bringmann et al., 2022]. 
4) A truly subquadratic-time algorithm and lower bound for Diameter-2 and Diameter-3 of rectangular boxes (of arbitrary dimension and sizes), respectively.

Cite As Get BibTex

Timothy M. Chan, Hsien-Chih Chang, Jie Gao, Sándor Kisfaludi-Bak, Hung Le, and Da Wei Zheng. Charting the Diameter Computation Landscape of Intersection Graphs in 3D and Above. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 29:1-29:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.29

Author Details

Timothy M. Chan
  • Siebel School of Computing and Data Science, University of Illinois Urbana-Champaign, IL, USA
Hsien-Chih Chang
  • Department of Computer Science, Dartmouth College, Hanover, NH, USA
Jie Gao
  • Department of Computer Science, Rutgers University, Piscataway, NJ, USA
Sándor Kisfaludi-Bak
  • Department of Computer Science, Aalto University, Espoo, Finland
Hung Le
  • Manning CICS, University of Massachusetts Amherst, Amherst, MA, USA
Da Wei Zheng
  • Institute of Science and Technology Austria, Klosterneuburg, Austria

Funding

  • Chan, Timothy M.: Supported by NSF grant CCF-2224271.
  • Chang, Hsien-Chih: Supported by NSF CAREER award CCF-2443017.
  • Gao, Jie: Supported by NSF DMS-2220271, DMS-2311064, IIS-2229876, CCF-2118953, CNS-2515159.
  • Kisfaludi-Bak, Sándor: Supported by the Research Council of Finland, Grant 363444.
  • Le, Hung: Supported by an NSF grant CCF-2517033 and an NSF CAREER Award CCF-2237288.
  • Zheng, Da Wei: This project has received funding from the Austrian Science Fund (FWF) grant DOI 10.55776/I5982. For open access purposes, the author has applied a CC BY public copyright license to any author-accepted manuscript version arising from this submission.

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