Document Open Access Logo

Enclosing Points with Geometric Objects

Authors Timothy M. Chan , Qizheng He , Jie Xue

PDF

File

Thumbnail PDF
PDF
LIPIcs.SoCG.2024.35.pdf
  • Filesize: 1.1 MB
  • 15 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
Keywords
  • obstacle placement
  • geometric optimization
  • approximation algorithms

Metrics

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

Abstract

Let X be a set of points in ℝ² and 𝒪 be a set of geometric objects in ℝ², where |X| + |𝒪| = n. We study the problem of computing a minimum subset 𝒪^* ⊆ 𝒪 that encloses all points in X. Here a point x ∈ X is enclosed by 𝒪^* if it lies in a bounded connected component of ℝ²∖(⋃_{O ∈ 𝒪^*} O). We propose two algorithmic frameworks to design polynomial-time approximation algorithms for the problem. The first framework is based on sparsification and min-cut, which results in O(1)-approximation algorithms for unit disks, unit squares, etc. The second framework is based on LP rounding, which results in an O(α(n)log n)-approximation algorithm for segments, where α(n) is the inverse Ackermann function, and an O(log n)-approximation algorithm for disks.

Cite As Get BibTex

Timothy M. Chan, Qizheng He, and Jie Xue. Enclosing Points with Geometric Objects. In 40th International Symposium on Computational Geometry (SoCG 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 293, pp. 35:1-35:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.SoCG.2024.35

Author Details

Timothy M. Chan
  • Department of Computer Science, University of Illinois Urbana-Champaign, IL, USA
Qizheng He
  • Department of Computer Science, University of Illinois Urbana-Champaign, IL, USA
Jie Xue
  • Department of Computer Science, New York University Shanghai, China

References

  1. Pankaj K. Agarwal and Micha Sharir. Davenport-Schinzel sequences and their geometric applications. In Handbook of Computational Geometry, pages 1-47. North Holland / Elsevier, 2000. Google Scholar
  2. Helmut Alt, Sergio Cabello, Panos Giannopoulos, and Christian Knauer. Minimum cell connection in line segment arrangements. International Journal of Computational Geometry & Applications, 27(03):159-176, 2017. Google Scholar
  3. Takao Asano, Tetsuo Asano, Leonidas Guibas, John Hershberger, and Hiroshi Imai. Visibility of disjoint polygons. Algorithmica, 1:49-63, 1986. Google Scholar
  4. Sayan Bandyapadhyay, Neeraj Kumar, Subhash Suri, and Kasturi R. Varadarajan. Improved approximation bounds for the minimum constraint removal problem. Comput. Geom., 90:101650, 2020. Google Scholar
  5. Sergey Bereg and David G. Kirkpatrick. Approximating barrier resilience in wireless sensor networks. In Algorithmic Aspects of Wireless Sensor Networks, 5th International Workshop (ALGOSENSORS), volume 5804 of Lecture Notes in Computer Science, pages 29-40. Springer, 2009. Google Scholar
  6. Sergio Cabello and Panos Giannopoulos. The complexity of separating points in the plane. Algorithmica, 74(2):643-663, 2016. Google Scholar
  7. David Yu Cheng Chan and David G. Kirkpatrick. Approximating barrier resilience for arrangements of non-identical disk sensors. In Algorithms for Sensor Systems, 8th International Symposium on Algorithms for Sensor Systems, Wireless Ad Hoc Networks and Autonomous Mobile Entities (ALGOSENSORS), volume 7718 of Lecture Notes in Computer Science, pages 42-53. Springer, 2012. Google Scholar
  8. Eduard Eiben, Jonathan Gemmell, Iyad A. Kanj, and Andrew Youngdahl. Improved results for minimum constraint removal. In Proceedings of the 32nd AAAI Conference on Artificial Intelligence, pages 6477-6484. AAAI Press, 2018. Google Scholar
  9. Lawrence H. Erickson and Steven M. LaValle. A simple, but NP-hard, motion planning problem. In Proceedings of the 27th AAAI Conference on Artificial Intelligence. AAAI Press, 2013. Google Scholar
  10. Subir Kumar Ghosh and David M Mount. An output-sensitive algorithm for computing visibility graphs. SIAM Journal on Computing, 20(5):888-910, 1991. Google Scholar
  11. Matt Gibson, Gaurav Kanade, Rainer Penninger, Kasturi R. Varadarajan, and Ivo Vigan. On isolating points using unit disks. J. Comput. Geom., 7(1):540-557, 2016. Google Scholar
  12. Leonidas J. Guibas, Micha Sharir, and Shmuel Sifrony. On the general motion-planning problem with two degrees of freedom. Discret. Comput. Geom., 4:491-521, 1989. Google Scholar
  13. John Hershberger and Subhash Suri. An optimal algorithm for euclidean shortest paths in the plane. SIAM Journal on Computing, 28(6):2215-2256, 1999. Google Scholar
  14. Sanjiv Kapoor and SN Maheshwari. Efficient algorithms for euclidean shortest path and visibility problems with polygonal obstacles. In Proceedings of the fourth annual symposium on computational geometry, pages 172-182, 1988. Google Scholar
  15. Klara Kedem, Ron Livne, János Pach, and Micha Sharir. On the union of jordan regions and collision-free translational motion amidst polygonal obstacles. Discrete & Computational Geometry, 1:59-70, 1986. Google Scholar
  16. Matias Korman, Maarten Löffler, Rodrigo I Silveira, and Darren Strash. On the complexity of barrier resilience for fat regions and bounded ply. Computational Geometry, 72:34-51, 2018. Google Scholar
  17. Neeraj Kumar, Daniel Lokshtanov, Saket Saurabh, and Subhash Suri. A constant factor approximation for navigating through connected obstacles in the plane. In Proceedings of the 32nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 822-839. SIAM, 2021. Google Scholar
  18. Neeraj Kumar, Daniel Lokshtanov, Saket Saurabh, Subhash Suri, and Jie Xue. Point separation and obstacle removal by finding and hitting odd cycles. In Proceedings of the 38th Symposium on Computational Geometry (SoCG), volume 224 of LIPIcs, pages 52:1-52:14, 2022. Google Scholar
  19. Santosh Kumar, Ten-Hwang Lai, and Anish Arora. Barrier coverage with wireless sensors. Wirel. Networks, 13(6):817-834, 2007. Google Scholar
  20. Joseph SB Mitchell. Shortest paths among obstacles in the plane. In Proceedings of the ninth annual symposium on Computational geometry, pages 308-317, 1993. Google Scholar
  21. Ricky Pollack, Micha Sharir, and Shmuel Sifrony. Separating two simple polygons by a sequence of translations. Discrete & Computational Geometry, 3:123-136, 1988. Google Scholar
  22. Prabhakar Raghavan. Probabilistic construction of deterministic algorithms: Approximating packing integer programs. J. Comput. Syst. Sci., 37(2):130-143, 1988. Google Scholar
  23. Prabhakar Raghavan and Clark D. Thompson. Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Comb., 7(4):365-374, 1987. Google Scholar
  24. M. Shimrat. Algorithm 112: Position of point relative to polygon. Commun. ACM, 5(8):434, 1962. Google Scholar
  25. Kuan-Chieh Robert Tseng. Resilience of wireless sensor networks. PhD thesis, University of British Columbia, 2011. Google Scholar
  26. Kuan-Chieh Robert Tseng and David G. Kirkpatrick. On barrier resilience of sensor networks. In Algorithms for Sensor Systems - 7th International Symposium on Algorithms for Sensor Systems, Wireless Ad Hoc Networks and Autonomous Mobile Entities (ALGOSENSORS), volume 7111 of Lecture Notes in Computer Science, pages 130-144. Springer, 2011. Google Scholar
  27. Vijay V. Vazirani. Approximation Algorithms. Springer, 2001. Google Scholar
  28. Haitao Wang. A new algorithm for euclidean shortest paths in the plane. Journal of the ACM, 70(2):1-62, 2023. Google Scholar
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact