Document Open Access Logo

Lower Bounds for Intersection Reporting Among Flat Objects

Authors Peyman Afshani, Pingan Cheng

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2023.3.pdf
  • Filesize: 0.73 MB
  • 16 pages

Document Identifiers

Author Details

Peyman Afshani
  • Aarhus University, Denmark
Pingan Cheng
  • Aarhus University, Denmark

Funding

Supported by DFF (Det Frie Forskningsråd) of Danish Council for Independent Research under grant ID DFF-7014-00404.

Cite AsGet BibTex

Peyman Afshani and Pingan Cheng. Lower Bounds for Intersection Reporting Among Flat Objects. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 3:1-3:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.SoCG.2023.3

Abstract

Recently, Ezra and Sharir [Esther Ezra and Micha Sharir, 2022] showed an O(n^{3/2+σ}) space and O(n^{1/2+σ}) query time data structure for ray shooting among triangles in ℝ³. This improves the upper bound given by the classical S(n)Q(n)⁴ = O(n^{4+σ}) space-time tradeoff for the first time in almost 25 years and in fact lies on the tradeoff curve of S(n)Q(n)³ = O(n^{3+σ}). However, it seems difficult to apply their techniques beyond this specific space and time combination. This pheonomenon appears persistently in almost all recent advances of flat object intersection searching, e.g., line-tetrahedron intersection in ℝ⁴ [Esther Ezra and Micha Sharir, 2022], triangle-triangle intersection in ℝ⁴ [Esther Ezra and Micha Sharir, 2022], or even among flat semialgebraic objects [Agarwal et al., 2022]. We give a timely explanation to this phenomenon from a lower bound perspective. We prove that given a set 𝒮 of (d-1)-dimensional simplicies in ℝ^d, any data structure that can report all intersections with a query line in small (n^o(1)) query time must use Ω(n^{2(d-1)-o(1)}) space. This dashes the hope of any significant improvement to the tradeoff curves for small query time and almost matches the classical upper bound. We also obtain an almost matching space lower bound of Ω(n^{6-o(1)}) for triangle-triangle intersection reporting in ℝ⁴ when the query time is small. Along the way, we further develop the previous lower bound techniques by Afshani and Cheng [Afshani and Cheng, 2021; Afshani and Cheng, 2022].

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Computational Geometry
  • Intersection Searching
  • Data Structure Lower Bounds

Metrics

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

Related Versions

References

  1. Peyman Afshani. Improved pointer machine and I/O lower bounds for simplex range reporting and related problems. In Proceedings of the Twenty-Eighth Annual Symposium on Computational Geometry, SoCG '12, pages 339-346, New York, NY, USA, 2012. Association for Computing Machinery. URL: https://doi.org/10.1145/2261250.2261301.
  2. Peyman Afshani and Pingan Cheng. Lower bounds for semialgebraic range searching and stabbing problems. In 37th International Symposium on Computational Geometry, volume 189 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 8, 15. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2021. Google Scholar
  3. Peyman Afshani and Pingan Cheng. On semialgebraic range reporting. In 38th International Symposium on Computational Geometry, volume 224 of LIPIcs. Leibniz Int. Proc. Inform., pages Paper No. 3, 14. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2022. URL: https://doi.org/10.4230/lipics.socg.2022.3.
  4. Pankaj K. Agarwal. Simplex range searching and its variants: a review. In A journey through discrete mathematics, pages 1-30. Springer, Cham, 2017. Google Scholar
  5. Pankaj K. Agarwal, Boris Aronov, Esther Ezra, Matthew J. Katz, and Micha Sharir. Intersection queries for flat semi-algebraic objects in three dimensions and related problems. In 38th International Symposium on Computational Geometry, volume 224 of LIPIcs. Leibniz Int. Proc. Inform., pages Paper No. 4, 14. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2022. URL: https://doi.org/10.4230/lipics.socg.2022.4.
  6. Pankaj K. Agarwal, Boris Aronov, Esther Ezra, and Joshua Zahl. Efficient algorithm for generalized polynomial partitioning and its applications. SIAM J. Comput., 50(2):760-787, 2021. URL: https://doi.org/10.1137/19M1268550.
  7. Pankaj K. Agarwal and Jirí Matousek. On range searching with semialgebraic sets. Discret. Comput. Geom., 11:393-418, 1994. URL: https://doi.org/10.1007/BF02574015.
  8. Pankaj K. Agarwal and Jiří Matoušek. Ray shooting and parametric search. SIAM J. Comput., 22(4):794-806, 1993. URL: https://doi.org/10.1137/0222051.
  9. Pankaj K. Agarwal, Jiří Matoušek, and Micha Sharir. On range searching with semialgebraic sets. II. SIAM J. Comput., 42(6):2039-2062, 2013. URL: https://doi.org/10.1137/120890855.
  10. Pankaj K. Agarwal and Micha Sharir. Ray shooting amidst convex polyhedra and polyhedral terrains in three dimensions. SIAM J. Comput., 25(1):100-116, 1996. URL: https://doi.org/10.1137/S0097539793244368.
  11. Boris Aronov, Mark de Berg, and Chris Gray. Ray shooting and intersection searching amidst fat convex polyhedra in 3-space. Comput. Geom., 41(1-2):68-76, 2008. URL: https://doi.org/10.1016/j.comgeo.2007.10.006.
  12. Timothy M. Chan. Optimal partition trees. Discrete Comput. Geom., 47(4):661-690, 2012. URL: https://doi.org/10.1007/s00454-012-9410-z.
  13. Bernard Chazelle. Lower bounds on the complexity of polytope range searching. J. Amer. Math. Soc., 2(4):637-666, 1989. URL: https://doi.org/10.2307/1990891.
  14. Bernard Chazelle. Lower bounds for orthogonal range searching. I. The reporting case. J. Assoc. Comput. Mach., 37(2):200-212, 1990. URL: https://doi.org/10.1145/77600.77614.
  15. Bernard Chazelle. Cutting hyperplanes for divide-and-conquer. Discrete Comput. Geom., 9(2):145-158, December 1993. URL: https://doi.org/10.1007/BF02189314.
  16. Bernard Chazelle and Burton Rosenberg. Simplex range reporting on a pointer machine. Comput. Geom., 5(5):237-247, 1996. URL: https://doi.org/10.1016/0925-7721(95)00002-X.
  17. M. de Berg, D. Halperin, M. Overmars, J. Snoeyink, and M. van Kreveld. Efficient ray shooting and hidden surface removal. Algorithmica, 12(1):30-53, 1994. URL: https://doi.org/10.1007/BF01377182.
  18. Mark de Berg and Chris Gray. Vertical ray shooting and computing depth orders for fat objects. SIAM J. Comput., 38(1):257-275, 2008. URL: https://doi.org/10.1137/060672261.
  19. Esther Ezra and Micha Sharir. Intersection searching amid tetrahedra in four dimensions. CoRR, abs/2208.06703, 2022. URL: https://doi.org/10.48550/arXiv.2208.06703.
  20. Esther Ezra and Micha Sharir. On ray shooting for triangles in 3-space and related problems. SIAM J. Comput., 51(4):1065-1095, 2022. URL: https://doi.org/10.1137/21M1408245.
  21. Larry Guth. Polynomial partitioning for a set of varieties. Math. Proc. Cambridge Philos. Soc., 159(3):459-469, 2015. URL: https://doi.org/10.1017/S0305004115000468.
  22. Larry Guth and Nets Hawk Katz. On the Erdős distinct distances problem in the plane. Ann. of Math. (2), 181(1):155-190, 2015. URL: https://doi.org/10.4007/annals.2015.181.1.2.
  23. Jiří Matoušek. Range searching with efficient hierarchical cuttings. Discrete Comput. Geom., 10(2):157-182, 1993. URL: https://doi.org/10.1007/BF02573972.
  24. Jiří Matoušek and Zuzana Patáková. Multilevel polynomial partitions and simplified range searching. Discrete Comput. Geom., 54(1):22-41, 2015. URL: https://doi.org/10.1007/s00454-015-9701-2.
  25. Jiří Matoušek and Otfried Schwarzkopf. On ray shooting in convex polytopes. Discrete Comput. Geom., 10(2):215-232, 1993. URL: https://doi.org/10.1007/BF02573975.
  26. M. Pellegrini. Ray shooting on triangles in 3-space. Algorithmica, 9(5):471-494, 1993. URL: https://doi.org/10.1007/BF01187036.
  27. Marco Pellegrini. Stabbing and ray shooting in 3 dimensional space. In Raimund Seidel, editor, Proceedings of the Sixth Annual Symposium on Computational Geometry, Berkeley, CA, USA, June 6-8, 1990, pages 177-186. ACM, 1990. URL: https://doi.org/10.1145/98524.98563.
  28. Marco Pellegrini. Ray shooting and lines in space. In Handbook of discrete and computational geometry (3rd Edition), CRC Press Ser. Discrete Math. Appl., pages 1093-1112. CRC, Boca Raton, FL, 2017. Google Scholar
  29. Edgar A. Ramos. On range reporting, ray shooting and k-level construction. In Proceedings of the Fifteenth Annual Symposium on Computational Geometry (Miami Beach, FL, 1999), pages 390-399. ACM, New York, 1999. URL: https://doi.org/10.1145/304893.304993.
  30. Micha Sharir and Hayim Shaul. Ray shooting and stone throwing with near-linear storage. Comput. Geom., 30(3):239-252, 2005. URL: https://doi.org/10.1016/j.comgeo.2004.10.001.
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