Document Open Access Logo

Geometric (1+ε)-Spanners with Few Crossings

Authors Kelvin Luu , Csaba D. Tóth

PDF

File

Thumbnail PDF
PDF
LIPIcs.GD.2026.36.pdf
  • Filesize: 1.24 MB
  • 18 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Sparsification and spanners
  • Theory of computation → Routing and network design problems
  • Mathematics of computing → Graphs and surfaces
Keywords
  • crossings
  • spanners
  • empty region property

Metrics

Abstract

For n points in the plane and an ε > 0, we construct a (1+ε)-spanner with O(n/ε) edges in which every edge has Õ(1/ε³) crossings, hence the total number of crossings is Õ(n/ε⁴), furthermore the ratio between the lengths of any two crossing edges is O(1/ε²). Our spanner construction substantially improves on the previous upper bound for the number of crossings in a (1+ε)-spanner, and it is the first spanner construction that ensures O(1) crossings per edge for any constant ε > 0. In contrast, we construct: n points in the plane for which every (1+ε)-spanner has Ω(n/ε³) crossings, n points for which every (1+ε)-spanner has an edge with Ω(1/ε^{5/2}) crossings, and 4 points for which every (1+ε)-spanner contains two crossing edges where one is Ω(1/ε) times longer than the other.

Cite As Get BibTex

Kelvin Luu and Csaba D. Tóth. Geometric (1+ε)-Spanners with Few Crossings. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 36:1-36:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.GD.2026.36

Author Details

Kelvin Luu
  • Department of Mathematics, California State University Northridge, Los Angeles, CA, USA
Csaba D. Tóth
  • Department of Mathematics, California State University Northridge, Los Angeles, CA, USA
  • Department of Computer Science, Tufts University, Medford, MA, USA

Funding

Research supported in part by the NSF award DMS-2154347.

References

  1. Mohammad Ali Abam and Sariel Har-Peled. New constructions of SSPDs and their applications. Comput. Geom., 45(5-6):200-214, 2012. URL: https://doi.org/10.1016/J.COMGEO.2011.12.003.
  2. Bernardo M. Ábrego, Ruy Fabila-Monroy, Silvia Fernández-Merchant, David Flores-Peñaloza, Ferran Hurtado, Vera Sacristán, and Maria Saumell. On crossing numbers of geometric proximity graphs. Comput. Geom., 44(4):216-233, 2011. URL: https://doi.org/10.1016/J.COMGEO.2010.11.003.
  3. Ingo Althöfer, Gautam Das, David Dobkin, Deborah Joseph, and José Soares. On sparse spanners of weighted graphs. Discrete & Computational Geometry, 9:81-100, 1993. URL: https://doi.org/10.1007/BF02189308.
  4. Sunil Arya, Gautam Das, David M. Mount, Jeffrey S. Salowe, and Michiel H. M. Smid. Euclidean spanners: short, thin, and lanky. In Proc. 27th ACM Symposium on Theory of Computing (STOC), pages 489-498, 1995. URL: https://doi.org/10.1145/225058.225191.
  5. Sunil Arya and Michiel H. M. Smid. Efficient construction of a bounded-degree spanner with low weight. Algorithmica, 17(1):33-54, 1997. URL: https://doi.org/10.1007/BF02523237.
  6. Sujoy Bhore, Sándor Kisfaludi-Bak, Lazar Milenkovic, Csaba D. Tóth, Karol Węgrzycki, and Sampson Wong. Euclidean noncrossing Steiner spanners of nearly optimal sparsity. In Proc. 42nd International Symposium on Computational Geometry (SoCG), volume 367 of LIPIcs, pages 15:1-15:18. Schloss Dagstuhl, 2026. URL: https://doi.org/10.4230/LIPIcs.SOCG.2026.15.
  7. Ahmad Biniaz, Mahdi Amani, Anil Maheshwari, Michiel H. M. Smid, Prosenjit Bose, and Jean-Lou De Carufel. A plane 1.88-spanner for points in convex position. J. Comput. Geom., 7(1):520-539, 2016. URL: https://doi.org/10.20382/JOCG.V7I1A21.
  8. Prosenjit Bose, Jean-Lou De Carufel, Pat Morin, André van Renssen, and Sander Verdonschot. Towards tight bounds on theta-graphs: More is not always better. Theor. Comput. Sci., 616:70-93, 2016. URL: https://doi.org/10.1016/J.TCS.2015.12.017.
  9. Prosenjit Bose, Luc Devroye, William S. Evans, and David G. Kirkpatrick. On the spanning ratio of Gabriel graphs and beta-skeletons. SIAM J. Discret. Math., 20(2):412-427, 2006. URL: https://doi.org/10.1137/S0895480197318088.
  10. Prosenjit Bose, Joachim Gudmundsson, and Pat Morin. Ordered theta graphs. Comput. Geom., 28(1):11-18, 2004. URL: https://doi.org/10.1016/J.COMGEO.2004.01.003.
  11. Prosenjit Bose and Michiel H. M. Smid. On plane geometric spanners: A survey and open problems. Comput. Geom., 46(7):818-830, 2013. URL: https://doi.org/10.1016/J.COMGEO.2013.04.002.
  12. Paul B. Callahan and S. Rao Kosaraju. A decomposition of multidimensional point sets with applications to k-nearest-neighbors and n-body potential fields. J. ACM, 42(1):67-90, 1995. URL: https://doi.org/10.1145/200836.200853.
  13. Jean Cardinal, Sébastien Collette, and Stefan Langerman. Empty region graphs. Comput. Geom., 42(3):183-195, 2009. URL: https://doi.org/10.1016/J.COMGEO.2008.09.003.
  14. Paz Carmi and Lilach Chaitman-Yerushalmi. Minimum weight Euclidean t-spanner is NP-hard. J. Discrete Algorithms, 22:30-42, 2013. URL: https://doi.org/10.1016/J.JDA.2013.06.010.
  15. Timothy M. Chan, Sariel Har-Peled, and Mitchell Jones. On locality-sensitive orderings and their applications. SIAM J. Comput., 49(3):583-600, 2020. URL: https://doi.org/10.1137/19M1246493.
  16. L. Paul Chew. There is a planar graph almost as good as the complete graph. In Proc. 2nd ACM Symposium on Computational Geometry (SoCG), pages 169-177, 1986. URL: https://doi.org/10.1145/10515.10534.
  17. Paul Chew. There are planar graphs almost as good as the complete graph. J. Comput. Syst. Sci., 39(2):205-219, 1989. URL: https://doi.org/10.1016/0022-0000(89)90044-5.
  18. Kenneth L. Clarkson. Approximation algorithms for shortest path motion planning. In Proc. 19th ACM Symposium on Theory of Computing (STOC), pages 56-65, 1987. URL: https://doi.org/10.1145/28395.28402.
  19. Gautam Das and Giri Narasimhan. A fast algorithm for constructing sparse Euclidean spanners. Int. J. Comput. Geom. Appl., 7(4):297-315, 1997. URL: https://doi.org/10.1142/S0218195997000193.
  20. Adrian Dumitrescu and Anirban Ghosh. Lower bounds on the dilation of plane spanners. Int. J. Comput. Geom. Appl., 26(2):89-110, 2016. URL: https://doi.org/10.1142/S0218195916500059.
  21. Michael Elkin and Shay Solomon. Optimal Euclidean spanners: Really short, thin, and lanky. J. ACM, 62(5):35:1-35:45, 2015. URL: https://doi.org/10.1145/2819008.
  22. David Eppstein and Hadi Khodabandeh. On the edge crossings of the greedy spanner. In Proc. 37th Symposium on Computational Geometry (SoCG), LIPIcs, pages 33:1-33:17. Schloss Dagstuhl, 2021. URL: https://doi.org/10.4230/LIPIcs.SOCG.2021.33.
  23. Mohammad Farshi and Joachim Gudmundsson. Experimental study of geometric t-spanners. ACM J. Exp. Algorithmics, 14, 2009. URL: https://doi.org/10.1145/1498698.1564499.
  24. Zhimeng Gao and Sariel Har-Peled. Near-optimal Euclidean locality-sensitive orderings. J. Comput. Geom., 16(2):91-108, 2024. URL: https://doi.org/10.20382/JOCG.V16I2A3.
  25. Panos Giannopoulos, Rolf Klein, Christian Knauer, Martin Kutz, and Dániel Marx. Computing geometric minimum-dilation graphs is NP-hard. Int. J. Comput. Geom. Appl., 20(2):147-173, 2010. URL: https://doi.org/10.1142/S0218195910003244.
  26. Joachim Gudmundsson, Christos Levcopoulos, and Giri Narasimhan. Fast greedy algorithms for constructing sparse geometric spanners. SIAM J. Comput., 31(5):1479-1500, 2002. URL: https://doi.org/10.1137/S0097539700382947.
  27. J. Mark Keil. Approximating the complete Euclidean graph. In Proc. 1st Scandinavian Workshop on Algorithm Theory (SWAT), volume 318 of LNCS, pages 208-213. Springer, 1988. URL: https://doi.org/10.1007/3-540-19487-8_23.
  28. David G. Kirkpatrick and John D. Radke. A framework for computational morphology. In Godfried T. Toussaint, editor, Computational Geometry, volume 2 of Machine Intelligence and Pattern Recognition, pages 217-248. North-Holland, 1985. URL: https://doi.org/10.1016/B978-0-444-87806-9.50013-X.
  29. Hung Le and Cuong Than. Greedy spanners in Euclidean spaces admit sublinear separators. ACM Trans. Algorithms, 20(3):23, 2024. URL: https://doi.org/10.1145/3590771.
  30. Giri Narasimhan and Michiel Smid. Geometric Spanner Networks. Cambridge University Press, 2007. URL: https://doi.org/10.1017/cbo9780511546884.
  31. André van Renssen, Yuan Sha, Yucheng Sun, and Sampson Wong. The tight spanning ratio of the rectangle Delaunay triangulation. Algorithmica, 87(7):1060-1080, 2025. URL: https://doi.org/10.1007/S00453-025-01308-W.
  32. Jim Ruppert and Raimund Seidel. Approximating the d-dimensional complete Euclidean graph. In Proc. 3rd Canadian Conference on Computational Geometry (CCCG), pages 207-210, 1991. URL: https://cccg.ca/proceedings/1991/paper50.pdf.
  33. Michiel Smid. New references related to geometric spanner networks, 2026. , accessed in April, 2026. URL: https://people.scs.carleton.ca/~michiel/SpannerBook/newreferences.html.
  34. Csaba D. Tóth. Minimum weight Euclidean (1+ε)-spanners. Eur. J. Comb., 118:103927, 2024. URL: https://doi.org/10.1016/J.EJC.2024.103927.
  35. Weizhao Wang, Xiang-Yang Li, Yu Wang, and Wen-Zhan Song. The spanning ratios of β-skeleton. In Proc. 15th Canadian Conference on Computational Geometry (Halifax, NS), pages 35-38, 2003. URL: http://www.cccg.ca/proceedings/2003/7.pdf.
  36. Ge Xia. The stretch factor of the Delaunay triangulation is less than 1.998. SIAM J. Comput., 42(4):1620-1659, 2013. URL: https://doi.org/10.1137/110832458.
  37. Ge Xia and Liang Zhang. Toward the tight bound of the stretch factor of Delaunay triangulations. In Proc. 23rd Canadian Conference on Computational Geometry (Toronto, ON), 2011. URL: http://www.cccg.ca/proceedings/2011/papers/paper57.pdf.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2026 Schloss Dagstuhl – LZI GmbH Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact