Fast and Exact Convex Hull Simplification

Authors Georgiy Klimenko, Benjamin Raichel

Thumbnail PDF


  • Filesize: 0.71 MB
  • 17 pages

Document Identifiers

Author Details

Georgiy Klimenko
  • Department of Computer Science, University of Texas at Dallas, Richardson, TX, USA
Benjamin Raichel
  • Department of Computer Science, University of Texas at Dallas, Richardson, TX, USA


The authors thank Sariel Har-Peled for helpful discussions related to the paper.

Cite AsGet BibTex

Georgiy Klimenko and Benjamin Raichel. Fast and Exact Convex Hull Simplification. In 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 213, pp. 26:1-26:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


Given a point set P in the plane, we seek a subset Q ⊆ P, whose convex hull gives a smaller and thus simpler representation of the convex hull of P. Specifically, let cost(Q,P) denote the Hausdorff distance between the convex hulls CH(Q) and CH(P). Then given a value ε > 0 we seek the smallest subset Q ⊆ P such that cost(Q,P) ≤ ε. We also consider the dual version, where given an integer k, we seek the subset Q ⊆ P which minimizes cost(Q,P), such that |Q| ≤ k. For these problems, when P is in convex position, we respectively give an O(n log²n) time algorithm and an O(n log³n) time algorithm, where the latter running time holds with high probability. When there is no restriction on P, we show the problem can be reduced to APSP in an unweighted directed graph, yielding an O(n^2.5302) time algorithm when minimizing k and an O(min{n^2.5302, kn^2.376}) time algorithm when minimizing ε, using prior results for APSP. Finally, we show our near linear algorithms for convex position give 2-approximations for the general case.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Convex hull
  • coreset
  • exact algorithm


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


  1. Pankaj K. Agarwal, Sariel Har-Peled, and Kasturi R. Varadarajan. Approximating extent measures of points. J. ACM, 51(4):606-635, 2004. URL:
  2. Pankaj K. Agarwal and Kasturi R. Varadarajan. Efficient algorithms for approximating polygonal chains. Discret. Comput. Geom., 23(2):273-291, 2000. URL:
  3. Alok Aggarwal, Heather Booth, Joseph O'Rourke, Subhash Suri, and Chee-Keng Yap. Finding minimal convex nested polygons. Inf. Comput., 83(1):98-110, 1989. URL:
  4. Noga Alon, Zvi Galil, and Oded Margalit. On the exponent of the all pairs shortest path problem. J. Comput. Syst. Sci., 54(2):255-262, 1997. URL:
  5. Avrim Blum, Sariel Har-Peled, and Benjamin Raichel. Sparse approximation via generating point sets. ACM Trans. Algorithms, 15(3):32:1-32:16, 2019. URL:
  6. Don Coppersmith and Shmuel Winograd. Matrix multiplication via arithmetic progressions. J. Symb. Comput., 9(3):251-280, 1990. URL:
  7. Ovidiu Daescu, Ningfang Mi, Chan-Su Shin, and Alexander Wolff. Farthest-point queries with geometric and combinatorial constraints. Comput. Geom., 33(3):174-185, 2006. URL:
  8. François Le Gall. Faster algorithms for rectangular matrix multiplication. In 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 514-523, 2012. URL:
  9. Jacob E. Goodman, Joseph O'Rourke, and Csaba D. Toth. Handbook of Discrete and Computational Geometry, Third Edition. CRC Press, 2017. Google Scholar
  10. Jacob E. Goodman, Joseph O'Rourke, and Csaba D. Tóth, editors. Handbook of Discrete and Computational Geometry, Third Edition. Chapman and Hall/CRC, 2018. Google Scholar
  11. Sariel Har-Peled and Benjamin Raichel. The fréchet distance revisited and extended. ACM Trans. Algorithms, 10(1):3:1-3:22, 2014. URL:
  12. G. Klimenko and B. Raichel. Fast and exact convex hull simplification, October 2021. URL:
  13. Georgiy Klimenko, Benjamin Raichel, and Gregory Van Buskirk. Sparse convex hull coverage. In Canadian Conference on Computational Geometry (CCCG), pages 15-25, 2020. Google Scholar
  14. Mario Alberto López and Shlomo Reisner. Hausdorff approximation of convex polygons. Comput. Geom., 32(2):139-158, 2005. URL:
  15. Mees van de Kerkhof, Irina Kostitsyna, Maarten Löffler, Majid Mirzanezhad, and Carola Wenk. Global curve simplification. In 27th Annual European Symposium on Algorithms (ESA), volume 144 of LIPIcs, pages 67:1-67:14, 2019. URL:
  16. Ivor van der Hoog, Vahideh Keikha, Maarten Löffler, Ali Mohades, and Jérôme Urhausen. Maximum-area triangle in a convex polygon, revisited. Inf. Process. Lett., 161:105943, 2020. URL:
  17. Marc J. van Kreveld, Maarten Löffler, and Lionov Wiratma. On optimal polyline simplification using the hausdorff and fréchet distance. J. Comput. Geom., 11(1):1-25, 2020. URL:
  18. Yanhao Wang, Michael Mathioudakis, Yuchen Li, and Kian-Lee Tan. Minimum coresets for maxima representation of multidimensional data. In Proceedings of the Symposium on Principles of Database Systems (PODS), pages 138-152. ACM, 2021. URL:
  19. Uri Zwick. All pairs shortest paths using bridging sets and rectangular matrix multiplication. J. ACM, 49(3):289-317, 2002. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail