Document Open Access Logo

On the Fragile Complexity of Geometric Algorithms

Authors Boris Aronov , Mayank Goswami , John Iacono , Indu Ramesh

PDF

File

Thumbnail PDF
PDF
LIPIcs.SWAT.2026.2.pdf
  • Filesize: 0.83 MB
  • 18 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Fragile complexity
  • convex hull
  • maxima
  • closest pair
  • algorithmic complexity

Metrics

Abstract

Surprisingly, the question of bounding the maximum number of operations undergone by each individual element in an algorithm - known as the fragile complexity of the algorithm - has not received much attention. In a foundational paper, Afshani et al. (2019) developed the concept of fragility and explored classic problems such as sorting and selection from this perspective. Motivated by a suggestion for future research by Afshani et al., we initiate a study of fragile complexity in computational geometry. We obtain bounds on several time-honored questions in 2D such as computing the maxima, closest pair, convex hull, triangulation, and approximate Euclidean Minimum Spanning Tree (apx-EMST). Our algorithms for the maxima, convex hull, and triangulation problems are competitive with the classical algorithms in terms of worst-case runtime and guarantee polylogarithmic fragility. We present an O(nlog²n) time algorithm that returns a 1.0125-apx-EMST and achieves O(log² n) fragility, thus matching the best known performance up to polylogarithmic factors.

Cite As Get BibTex

Boris Aronov, Mayank Goswami, John Iacono, and Indu Ramesh. On the Fragile Complexity of Geometric Algorithms. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 2:1-2:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SWAT.2026.2

Author Details

Boris Aronov
  • Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA
Mayank Goswami
  • Department of Computer Science, Queens College CUNY, New York, NY, USA
John Iacono
  • Université libre de Bruxelles, Ixelles, Belgium
Indu Ramesh
  • Department of Computer Science and Engineering, Tandon School of Engineering, New York University, Brooklyn, NY, USA

Funding

  • Aronov, Boris: Partially supported by NSF Grant CCF-20-08551.
  • Goswami, Mayank: Supported by National Science Foundation CCF-2503086. Part of this work was performed at the AlgoPARC Workshop on Parallel Algorithms and Data Structures at the University of Hawai'i at Mānoa, USA, with the support of the National Science Foundation (USA) under Grant No. 2452276.
  • Iacono, John: Research supported by the Fonds de la Recherche Scientifique - FNRS (Belgium). Part of this work was performed at the AlgoPARC Workshop on Parallel Algorithms and Data Structures at the University of Hawai'i at Mānoa, USA, with the support of the National Science Foundation (USA) under Grant No. 2452276.
  • Ramesh, Indu: Supported by NSF Grant CCF-20-08551.

References

  1. Peyman Afshani, Rolf Fagerberg, David Hammer, Riko Jacob, Irina Kostitsyna, Ulrich Meyer, Manuel Penschuck, and Nodari Sitchinava. Fragile Complexity of Comparison-Based Algorithms. In 27th Annual European Symposium on Algorithms (ESA 2019), volume 144 of Leibniz International Proceedings in Informatics (LIPIcs), pages 2:1-2:19, Dagstuhl, Germany, 2019. Schloss Dagstuhl - Leibniz-Zentrum für Informatik. URL: https://doi.org/10.4230/LIPIcs.ESA.2019.2.
  2. Pankaj K Agarwal, Herbert Edelsbrunner, Otfried Schwarzkopf, and Emo Welzl. Euclidean minimum spanning trees and bichromatic closest pairs. In Proceedings of the Sixth Annual Symposium on Computational Geometry, pages 203-210, 1990. URL: https://doi.org/10.1145/98524.98567.
  3. Miklós Ajtai, János Komlós, and Endre Szemerédi. An O(n log n) sorting network. In Proceedings of the 15th Annual ACM Symposium on Theory of Computing, 25-27 April, 1983, Boston, Massachusetts, USA, pages 1-9. ACM, 1983. URL: https://doi.org/10.1145/800061.808726.
  4. Michael Ben-Or. Lower bounds for algebraic computation trees. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, pages 80-86, 1983. Google Scholar
  5. Prosenjit Bose, Pilar Cano, Rolf Fagerberg, John Iacono, Riko Jacob, and Stefan Langerman. Fragile complexity of adaptive algorithms. In International Conference on Algorithms and Complexity, pages 144-157. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-75242-2_10.
  6. Prosenjit Bose, Anil Maheshwari, Pat Morin, Jason Morrison, Michiel Smid, and Jan Vahrenhold. Space-efficient geometric divide-and-conquer algorithms. Computational Geometry, 37(3):209-227, 2007. URL: https://doi.org/10.1016/J.COMGEO.2006.03.006.
  7. Paul B Callahan. Dealing with higher dimensions: the well-separated pair decomposition and its applications. The Johns Hopkins University, 1995. Google Scholar
  8. Thomas H Cormen, Charles E Leiserson, Ronald L Rivest, and Clifford Stein. Introduction to algorithms. MIT press, 2022. Google Scholar
  9. Karthik C.S. and Pasin Manurangsi. On closest pair in Euclidean metric: Monochromatic is as hard as bichromatic. Combinatorica, 40(4):539-573, 2020. URL: https://doi.org/10.1007/S00493-019-4113-1.
  10. Mark De Berg, Otfried Cheong, Marc Van Kreveld, and Mark Overmars. Computational Geometry: Algorithms and Applications. Springer, 2008. Google Scholar
  11. Jeff Erickson. Lecture notes for cs 497: Concrete models of computation, spring 2003. Google Scholar
  12. Ronald L. Graham. An efficient algorithm for determining the convex hull of a finite planar set. Info. Proc. Lett., 1:132-133, 1972. URL: https://doi.org/10.1016/0020-0190(72)90045-2.
  13. Ray A Jarvis. On the identification of the convex hull of a finite set of points in the plane. Information Processing Letters, 2(1):18-21, 1973. URL: https://doi.org/10.1016/0020-0190(73)90020-3.
  14. Rajesh Jayaram, Vahab Mirrokni, Shyam Narayanan, and Peilin Zhong. Massively parallel algorithms for high-dimensional euclidean minimum spanning tree. In Proceedings of the 2024 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3960-3996. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.139.
  15. David G Kirkpatrick and Raimund Seidel. Output-size sensitive algorithms for finding maximal vectors. In Proceedings of the First Annual Symposium on Computational Geometry, pages 89-96, 1985. URL: https://doi.org/10.1145/323233.323246.
  16. David G Kirkpatrick and Raimund Seidel. The ultimate planar convex hull algorithm? SIAM Journal on Computing, 15(1):287-299, 1986. URL: https://doi.org/10.1137/0215021.
  17. H. T. Kung, Fabrizio Luccio, and Franco P. Preparata. On finding the maxima of a set of vectors. J. ACM, 22(4):469-476, 1975. URL: https://doi.org/10.1145/321906.321910.
  18. Mark H Overmars and Jan Van Leeuwen. Maintenance of configurations in the plane. Journal of computer and System Sciences, 23(2):166-204, 1981. URL: https://doi.org/10.1016/0022-0000(81)90012-X.
  19. Franco P. Preparata and Se June Hong. Convex hulls of finite sets of points in two and three dimensions. Communications of the ACM, 20(2):87-93, 1977. URL: https://doi.org/10.1145/359423.359430.
  20. Franco P Preparata and Michael I Shamos. Computational Geometry: An Introduction. Springer Science & Business Media, 2012. Google Scholar
  21. Michael Ian Shamos and Dan Hoey. Closest-point problems. In 16th Annual Symposium on Foundations of Computer Science (sfcs 1975), pages 151-162. IEEE, 1975. URL: https://doi.org/10.1109/SFCS.1975.8.
  22. Andrew Chi-Chih Yao. On constructing minimum spanning trees in k-dimensional spaces and related problems. SIAM Journal on Computing, 11(4):721-736, 1982. URL: https://doi.org/10.1137/0211059.
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