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Online Hitting Sets for Disks of Bounded Radii

Authors Minati De , Satyam Singh , Csaba D. Tóth

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Online algorithms
Keywords
  • Geometric Hitting Set
  • Online Algorithm
  • Homothets
  • Disks

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Abstract

We present algorithms for the online minimum hitting set problem in geometric range spaces: Given a set P of n points in the plane and a sequence of geometric objects that arrive one-by-one, we need to maintain a hitting set at all times. For disks of radii in the interval [1,M], we present an O(log M log n)-competitive algorithm. This result generalizes from disks to positive homothets of any convex body in the plane with scaling factors in the interval [1,M]. As a main technical tool, we reduce the problem to the online hitting set problem for a finite subset of integer points and bottomless rectangles. Specifically, for a given N > 1, we present an O(log N)-competitive algorithm for the variant where P is a subset of an N× N section of the integer lattice, and the geometric objects are bottomless rectangles.

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Minati De, Satyam Singh, and Csaba D. Tóth. Online Hitting Sets for Disks of Bounded Radii. In 33rd Annual European Symposium on Algorithms (ESA 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 351, pp. 50:1-50:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ESA.2025.50

Author Details

Minati De
  • Dept. 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

  • De, Minati: Research on this paper was supported by SERB MATRICS Grant MTR/2021/ 000584.
  • 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.

References

  1. 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.
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  7. Minati De and Satyam Singh. Hitting geometric objects online via points in ℤ^d. In Proc. 28th Conf. Computing and Combinatorics (COCOON), volume 13595 of LNCS, pages 537-548. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-22105-7_48.
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