Document Open Access Logo

Covering Weighted Points Using Unit Squares

Authors Chaeyoon Chung , Jaegun Lee , Hee-Kap Ahn

PDF

File

Thumbnail PDF
PDF
LIPIcs.ISAAC.2025.21.pdf
  • Filesize: 0.93 MB
  • 14 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • Maximum coverage
  • Unit squares
  • Approximation algorithms

Metrics

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

Abstract

Given a set of n points in d-dimensional space, each assigned a positive weight, we study the problem of finding k axis-parallel unit hypercubes that maximize the total weight of the points contained in their union. In this paper, we present both exact and (1 - ε)-approximation algorithms for the case of k = 2. We present an exact algorithm that runs in O(n²) time in the plane, improving the previous O(n² log² n)-time result. This algorithm generalizes to higher dimensions and larger k in O(n^{dk/2}) time for fixed d and k. We also present a (1 - ε)-approximation algorithm that runs in O(n log min{n, 1/ε} + 1/ε³) time for k = 2 in the plane, improving the best known result. Our approximation algorithm also extends to higher dimensions.

Cite As Get BibTex

Chaeyoon Chung, Jaegun Lee, and Hee-Kap Ahn. Covering Weighted Points Using Unit Squares. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 21:1-21:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ISAAC.2025.21

Author Details

Chaeyoon Chung
  • Department of Computer Science and Engineering, Pohang University of Science and Technology, South Korea
Jaegun Lee
  • Department of Computer Science and Engineering, Pohang University of Science and Technology, South Korea
Hee-Kap Ahn
  • Department of Computer Science and Engineering, Graduate School of Artificial Intelligence, Pohang University of Science and Technology, South Korea

Funding

This work was partly supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government(MSIT) (RS-2023-00219980), and the Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government (MSIT) (No.RS-2019-II191906, Artificial Intelligence Graduate School Program (POSTECH)) and (No. 2017-0-00905, Software Star Lab (Optimal Data Structure and Algorithmic Applications in Dynamic Geometric Environment)).

References

  1. Michael Ben-Or. Lower bounds for algebraic computation trees. In Proceedings of 15th Annual ACM Symposium on Theory of Computing, pages 80-86, New York, NY, USA, 1983. Association for Computing Machinery. Google Scholar
  2. Sergio Cabello, J. Miguel Díaz-Báñez, Carlos Seara, J. Antoni Sellès, Jorge Urrutia, and Inmaculada Ventura. Covering point sets with two disjoint disks or squares. Computational Geometry, 40(3):195-206, 2008. URL: https://doi.org/10.1016/J.COMGEO.2007.10.001.
  3. Timothy M. Chan. Klee’s measure problem made easy. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pages 410-419, 2013. URL: https://doi.org/10.1109/FOCS.2013.51.
  4. Mark de Berg, Sergio Cabello, and Sariel Har-Peled. Covering many or few points with unit disks. Theory of Computing Systems, 45(3):446-469, 2009. URL: https://doi.org/10.1007/S00224-008-9135-9.
  5. Uriel Feige. A threshold of ln n for approximating set cover. Journal of the ACM, 45(4):634-652, 1998. URL: https://doi.org/10.1145/285055.285059.
  6. Dorit S. Hochbaum and Anu Pathria. Analysis of the greedy approach in problems of maximum k-coverage. Naval Research Logistics, 45(6):615-627, 1998. Google Scholar
  7. Hiroshi Imai and Takao Asano. Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane. Journal of Algorithms, 4(4):310-323, 1983. URL: https://doi.org/10.1016/0196-6774(83)90012-3.
  8. Kai Jin, Jian Li, Haitao Wang, Bowei Zhang, and Ningye Zhang. Near-linear time approximation schemes for geometric maximum coverage. Theoretical Computer Science, 725:64-78, 2018. URL: https://doi.org/10.1016/J.TCS.2017.11.026.
  9. Priya Ranjan Sinha Mahapatra, Partha P. Goswami, and Sandip Das. Placing two axis-parallel squares to maximize the number of enclosed points. International Journal of Computational Geometry & Applications, 25(04):263-282, 2015. URL: https://doi.org/10.1142/S0218195915500156.
  10. Nimrod Megiddo and Kenneth J. Supowit. On the complexity of some common geometric location problems. SIAM Journal on Computing, 13(1):182-196, 1984. URL: https://doi.org/10.1137/0213014.
  11. Ali Mostafavi and Ali Hamzeh. High-dimensional axis-aligned bounding box with outliers. In Proceedings of 34th Canadian Conference on Computational Geometry, pages 264-269, 2022. Google Scholar
  12. Subhas C. Nandy and Bhargab B. Bhattacharya. A unified algorithm for finding maximum and minimum object enclosing rectangles and cuboids. Computers & Mathematics with Applications, 29(8):45-61, 1995. Google Scholar
  13. George L. Nemhauser, Laurence Wolsey, and M.L. Fisher. An analysis of approximations for maximizing submodular set functions-I. Mathematical Programming, 14(1):265-294, 1978. URL: https://doi.org/10.1007/BF01588971.
  14. Yufei Tao, Xiaocheng Hu, Dong-Wan Choi, and Chin-Wan Chung. Approximate MaxRS in spatial databases. In Proceedings of the VLDB Endowment, pages 1546-1557. VLDB Endowment, 2013. URL: https://doi.org/10.14778/2536258.2536266.
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact