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Online Hitting Set for Axis-Aligned Squares

Authors Minati De , Satyam Singh , Csaba D. Tóth

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LIPIcs.SWAT.2026.16.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Online algorithms
Keywords
  • axis-aligned squares
  • hitting set
  • homothets of a polygon
  • online algorithm

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Abstract

Given a set P of n points in the plane and a sequence of axis-aligned squares that arrive in an online fashion, the online hitting set problem consists of maintaining, by adding new points from P if necessary, a hitting set H ⊆ P, which contains at least one point in every input square that has already arrived. We present an O(log n)-competitive deterministic algorithm for this problem. The competitive ratio is the best possible, apart from constant factors. In fact, this is the first O(log n)-competitive algorithm for the online hitting set problem that works for geometric objects of arbitrary sizes (i.e., unbounded scaling factors) in the plane. We further generalize this result to positive homothets of a polygon with k ≥ 3 vertices in the plane and provide an O(k²log n)-competitive algorithm.

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Minati De, Satyam Singh, and Csaba D. Tóth. Online Hitting Set for Axis-Aligned Squares. In 20th Scandinavian Symposium on Algorithm Theory (SWAT 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 370, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SWAT.2026.16

Author Details

Minati De
  • Department of Mathematics, Indian Institute of Technology Delhi, New Delhi, India
Satyam Singh
  • Department of Computer Science, Aalto University, Espoo, Finland
Csaba D. Tóth
  • Department of Mathematics, California State University Northridge, Los Angeles, CA, USA
  • Department of Computer Science, Tufts University, Medford, MA, USA

Funding

  • Singh, Satyam: Research on this paper was supported by the Research Council of Finland, Grant 363444.
  • Tóth, Csaba D.: Research on this paper was supported, in part, by the NSF award DMS-2154347.

Acknowledgements

We thank an anonymous reviewer for a suggestion that helped improve the competitive ratio in Theorem 1 from O(ϱ log n) to O((1+log ϱ)log n).

References

  1. Pankaj K. Agarwal, Kyle Fox, Kamesh Munagala, and Abhinandan Nath. Parallel algorithms for constructing range and nearest-neighbor searching data structures. In Proc. 35th ACM Symposium on Principles of Database Systems, (PODS), pages 429-440. ACM, 2016. URL: https://doi.org/10.1145/2902251.2902303.
  2. Shanli Alefkhani, Nima Khodaveisi, and Mathieu Mari. Online hitting set of d-dimensional fat objects. In Proc. 21st International Workshop on Approximation and Online Algorithms, (WAOA), volume 14297 of LNCS, pages 134-144. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-49815-2_10.
  3. Noga Alon, Baruch Awerbuch, Yossi Azar, Niv Buchbinder, and Joseph Naor. The online set cover problem. SIAM J. Comput., 39(2):361-370, 2009. URL: https://doi.org/10.1137/060661946.
  4. Sunil Arya, Theocharis Malamatos, and David M. Mount. Space-time tradeoffs for approximate nearest neighbor searching. J. ACM, 57(1):1:1-1:54, 2009. URL: https://doi.org/10.1145/1613676.1613677.
  5. Sunil Arya and David M. Mount. Approximate range searching. Comput. Geom., 17(3-4):135-152, 2000. URL: https://doi.org/10.1016/S0925-7721(00)00022-5.
  6. Sunil Arya, David M. Mount, Nathan S. Netanyahu, Ruth Silverman, and Angela Y. Wu. An optimal algorithm for approximate nearest neighbor searching fixed dimensions. J. ACM, 45(6):891-923, 1998. URL: https://doi.org/10.1145/293347.293348.
  7. Mark de Berg, Otfried Cheong, Marc J. van Kreveld, and Mark H. Overmars. Computational Geometry: Algorithms and Applications. Springer, 3rd edition, 2008. URL: https://doi.org/10.1007/978-3-540-77974-2.
  8. Mark de Berg, Haggai David, Matthew J. Katz, Mark H. Overmars, A. Frank van der Stappen, and Jules Vleugels. Guarding scenes against invasive hypercubes. Comput. Geom., 26(2):99-117, 2003. URL: https://doi.org/10.1016/S0925-7721(03)00016-6.
  9. Sujoy Bhore, Anupam Gupta, and Amit Kumar. Improved Online Hitting Set Algorithms for Structured and Geometric Set Systems. In 42nd International Symposium on Computational Geometry (SoCG 2026), volume 367 of Leibniz International Proceedings in Informatics (LIPIcs), pages 14:1-14:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2026. URL: https://doi.org/10.4230/LIPIcs.SoCG.2026.14.
  10. Srecko Brlek, Hugo Tremblay, Jérôme Tremblay, and Romaine Weber. Efficient computation of the outer hull of a discrete path. In Proc. 18th IAPR International Conference on Discrete Geometry for Computer Imagery (DGCI), volume 8668 of LNCS, pages 122-133. Springer, 2014. URL: https://doi.org/10.1007/978-3-319-09955-2_11.
  11. Timothy M. Chan. Approximate nearest neighbor queries revisited. Discret. Comput. Geom., 20(3):359-373, 1998. URL: https://doi.org/10.1007/PL00009390.
  12. Timothy M. Chan, Kasper Green Larsen, and Mihai Pătraşcu. Orthogonal range searching on the ram, revisited. In Proc. 27th Symposium on Computational Geometry, pages 1-10. ACM Press, 2011. URL: https://doi.org/10.1145/1998196.1998198.
  13. Timothy M. Chan and Konstantinos Tsakalidis. Dynamic orthogonal range searching on the RAM, revisited. J. Comput. Geom., 9(2):45-66, 2018. URL: https://doi.org/10.20382/JOCG.V9I2A5.
  14. Moses Charikar, Chandra Chekuri, Tomás Feder, and Rajeev Motwani. Incremental clustering and dynamic information retrieval. SIAM J. Comput., 33(6):1417-1440, 2004. URL: https://doi.org/10.1137/S0097539702418498.
  15. Bernard Chazelle. A functional approach to data structures and its use in multidimensional searching. SIAM J. Computing, 17(3):427-462, 1988. URL: https://doi.org/10.1137/0217026.
  16. Danny Z. Chen and Haitao Wang. Computing L₁ shortest paths among polygonal obstacles in the plane. Algorithmica, 81(6):2430-2483, 2019. URL: https://doi.org/10.1007/S00453-018-00540-X.
  17. Minati De, Saksham Jain, Sarat Varma Kallepalli, and Satyam Singh. Online geometric covering and piercing. Algorithmica, 86(9):2739-2765, 2024. URL: https://doi.org/10.1007/S00453-024-01244-1.
  18. Minati De, Ratnadip Mandal, and Satyam Singh. New lower bound and algorithms for online geometric hitting set problem. arXiv, 2024. URL: https://doi.org/10.48550/arXiv.2409.11166.
  19. Minati De and Satyam Singh. Online hitting of unit balls and hypercubes in ℝ^d using points from ℤ^d. Theor. Comput. Sci., 992:114452, 2024. URL: https://doi.org/10.1016/J.TCS.2024.114452.
  20. Minati De, Satyam Singh, and Csaba D. Tóth. Online hitting sets for disks of bounded radii. In Proc. 33rd European Symposium on Algorithms, (ESA), volume 351 of LIPIcs, pages 50:1-50:16. Schloss Dagstuhl, 2025. URL: https://doi.org/10.4230/LIPIcs.ESA.2025.50.
  21. Adrian Dumitrescu, Anirban Ghosh, and Csaba D. Tóth. Online unit covering in Euclidean space. Theor. Comput. Sci., 809:218-230, 2020. URL: https://doi.org/10.1016/J.TCS.2019.12.010.
  22. Adrian Dumitrescu and Csaba D. Tóth. Online unit clustering and unit covering in higher dimensions. Algorithmica, 84(5):1213-1231, 2022. URL: https://doi.org/10.1007/S00453-021-00916-6.
  23. Guy Even and Shakhar Smorodinsky. Hitting sets online and unique-max coloring. Discret. Appl. Math., 178:71-82, 2014. URL: https://doi.org/10.1016/J.DAM.2014.06.019.
  24. M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  25. Ziyue Huang and Ke Yi. Approximate range counting under differential privacy. In Proc. 37th International Symposium on Computational Geometry (SoCG), volume 189 of LIPIcs, pages 45:1-45:14. Schloss Dagstuhl, 2021. URL: https://doi.org/10.4230/LIPIcs.SoCG.2021.45.
  26. Arindam Khan, Aditya Lonkar, Saladi Rahul, Aditya Subramanian, and Andreas Wiese. Online and dynamic algorithms for geometric set cover and hitting set. In Proc. 39th Symposium on Computational Geometry (SoCG), volume 258 of LIPIcs, pages 46:1-46:17. Schloss Dagstuhl, 2023. See also https://arxiv.org/abs/2303.09524. URL: https://doi.org/10.4230/LIPIcs.SOCG.2023.46.
  27. Kurt Mehlhorn and Stefan Näher. Dynamic fractional cascading. Algorithmica, 5(2):215-241, 1990. URL: https://doi.org/10.1007/BF01840386.
  28. Christian Worm Mortensen. Fully dynamic orthogonal range reporting on RAM. SIAM J. Comput., 35(6):1494-1525, 2006. URL: https://doi.org/10.1137/S0097539703436722.
  29. Joseph O'Rourke. Art Gallery Theorems and Algorithms. Oxford University Press, 1987. URL: https://www.science.smith.edu/~jorourke/books/ArtGalleryTheorems/.
  30. Haitao Wang. A new algorithm for Euclidean shortest paths in the plane. In Proc. 53rd ACM SIGACT Symposium on Theory of Computing, (STOC), pages 975-988. ACM, 2021. URL: https://doi.org/10.1145/3406325.3451037.
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