Document Open Access Logo

Sweeping Arrangements of Non-Piercing Regions in the Plane

Authors Suryendu Dalal, Rahul Gangopadhyay, Rajiv Raman, Saurabh Ray

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2024.45.pdf
  • Filesize: 0.86 MB
  • 15 pages

Document Identifiers

Author Details

Suryendu Dalal
  • IIIT-Delhi, India
Rahul Gangopadhyay
  • Moscow Institute of Physics and Technology, Russia
Rajiv Raman
  • IIIT-Delhi, India
Saurabh Ray
  • NYU Abu Dhai, UAE

Funding

  • Gangopadhyay, Rahul: Rahul Gangopadhyay was supported by a grant for research centers in the field of artificial intelligence, provided by the Analytical Center for the Government of the Russian Federation in accordance with the subsidy agreement (agreement identifier 000000D730321P5Q0002) and the agreement with the Moscow Institute of Physics and Technology dated November 1, 2021 No. 70-2021-00138.

Acknowledgements

We would like to thank the anonymous reviewers for their constructive comments that improved the presentation of the paper.

Cite AsGet BibTex

Suryendu Dalal, Rahul Gangopadhyay, Rajiv Raman, and Saurabh Ray. Sweeping Arrangements of Non-Piercing Regions in the Plane. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 45:1-45:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.SoCG.2024.45

Abstract

Let Γ be a finite set of Jordan curves in the plane. For any curve γ ∈ Γ, we denote the bounded region enclosed by γ as γ̃. We say that Γ is a non-piercing family if for any two curves α , β ∈ Γ, α̃ ⧵ β̃ is a connected region. A non-piercing family of curves generalizes a family of 2-intersecting curves in which each pair of curves intersect in at most two points. Snoeyink and Hershberger ("Sweeping Arrangements of Curves", SoCG '89) proved that if we are given a family Γ of 2-intersecting curves and a sweep curve γ ∈ Γ, then the arrangement can be swept by γ while always maintaining the 2-intersecting property of the curves. We generalize the result of Snoeyink and Hershberger to the setting of non-piercing curves. We show that given an arrangement of non-piercing curves Γ, and a sweep curve γ ∈ Γ, the arrangement can be swept by γ so that the arrangement remains non-piercing throughout the process. We also give a shorter and simpler proof of the result of Snoeyink and Hershberger, and give an eclectic set of applications.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Sweeping
  • Pseudodisks
  • Discrete Geometry
  • Topology

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

References

  1. Eyal Ackerman, Balázs Keszegh, and Dömötör Pálvölgyi. Coloring hypergraphs defined by stabbed pseudo-disks and abab-free hypergraphs. SIAM Journal on Discrete Mathematics, 34(4):2250-2269, 2020. Google Scholar
  2. Eyal Ackerman, Balázs Keszegh, and Dömötör Pálvölgyi. Coloring delaunay-edges and their generalizations. Computational Geometry, 96:101745, 2021. Google Scholar
  3. Pankaj K Agarwal, Eran Nevo, János Pach, Rom Pinchasi, Micha Sharir, and Shakhar Smorodinsky. Lenses in arrangements of pseudo-circles and their applications. Journal of the ACM (JACM), 51(2):139-186, 2004. Google Scholar
  4. Bentley and Ottmann. Algorithms for reporting and counting geometric intersections. IEEE Transactions on computers, 100(9):643-647, 1979. Google Scholar
  5. Sarit Buzaglo, Rom Holzman, and Rom Pinchasi. On s-intersecting curves and related problems. In Proceedings of the twenty-fourth annual Symposium on Computational geometry, pages 79-84, 2008. Google Scholar
  6. Timothy M. Chan and Elyot Grant. Exact algorithms and apx-hardness results for geometric packing and covering problems. Comput. Geom., 47(2):112-124, 2014. URL: https://doi.org/10.1016/J.COMGEO.2012.04.001.
  7. Timothy M. Chan, Elyot Grant, Jochen Könemann, and Malcolm Sharpe. Weighted capacitated, priority, and geometric set cover via improved quasi-uniform sampling. In Yuval Rabani, editor, Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, January 17-19, 2012, pages 1576-1585. SIAM, 2012. URL: https://doi.org/10.1137/1.9781611973099.125.
  8. Bernard Chazelle and Herbert Edelsbrunner. An optimal algorithm for intersecting line segments in the plane. Journal of the ACM (JACM), 39(1):1-54, 1992. Google Scholar
  9. Man-Kwun Chiu, Stefan Felsner, Manfred Scheucher, Felix Schröder, Raphael Steiner, and Birgit Vogtenhuber. Coloring circle arrangements: New 4-chromatic planar graphs. European Journal of Combinatorics, page 103839, 2023. Google Scholar
  10. Mark de Berg, Otfried Cheong, Marc J. van Kreveld, and Mark H. Overmars. Computational geometry: algorithms and applications, 3rd Edition. Springer, 2008. URL: https://www.worldcat.org/oclc/227584184.
  11. Herbert Edelsbrunner. Algorithms in combinatorial geometry, volume 10. Springer Science & Business Media, 1987. Google Scholar
  12. Herbert Edelsbrunner and Leonidas J. Guibas. Topologically sweeping an arrangement. Journal of Computer and System Sciences, 38(1):165-194, 1989. URL: https://doi.org/10.1016/0022-0000(89)90038-X.
  13. Stefan Felsner and Klaus Kriegel. Triangles in euclidean arrangements. Discrete & Computational Geometry, 22:429-438, 1999. Google Scholar
  14. Sariel Har-Peled and Kent Quanrud. Approximation algorithms for polynomial-expansion and low-density graphs. SIAM J. Comput., 46(6):1712-1744, 2017. URL: https://doi.org/10.1137/16M1079336.
  15. Chaya Keller, Balázs Keszegh, and Dömötör Pálvölgyi. On the number of hyperedges in the hypergraph of lines and pseudo-discs. Electron. J. Comb., 29(3), 2022. URL: https://doi.org/10.37236/10424.
  16. Rom Pinchasi. A finite family of pseudodiscs must include a "small" pseudodisc. SIAM J. Discret. Math., 28(4):1930-1934, 2014. URL: https://doi.org/10.1137/130949750.
  17. Franco P Preparata and Michael I Shamos. Computational geometry: an introduction. Springer Science & Business Media, 2012. Google Scholar
  18. Rajiv Raman and Saurabh Ray. Constructing planar support for non-piercing regions. Discret. Comput. Geom., 64(3):1098-1122, 2020. URL: https://doi.org/10.1007/S00454-020-00216-W.
  19. Rajiv Raman and Saurabh Ray. On the geometric set multicover problem. Discret. Comput. Geom., 68(2):566-591, 2022. URL: https://doi.org/10.1007/S00454-022-00402-Y.
  20. Manfred Scheucher. Points, lines, and circles:: some contributions to combinatorial geometry. PhD thesis, Dissertation, Berlin, Technische Universität Berlin, 2019, 2020. Google Scholar
  21. Jack Snoeyink and John Hershberger. Sweeping arrangements of curves. In Kurt Mehlhorn, editor, Proceedings of the Fifth Annual Symposium on Computational Geometry, Saarbrücken, Germany, June 5-7, 1989, pages 354-363. ACM, 1989. URL: https://doi.org/10.1145/73833.73872.
  22. Kasturi R. Varadarajan. Weighted geometric set cover via quasi-uniform sampling. In Leonard J. Schulman, editor, Proceedings of the 42nd ACM Symposium on Theory of Computing, STOC 2010, Cambridge, Massachusetts, USA, 5-8 June 2010, pages 641-648. ACM, 2010. URL: https://doi.org/10.1145/1806689.1806777.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact