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Unweighted Geometric Hitting Set for Line-Constrained Disks and Related Problems

Authors Gang Liu, Haitao Wang

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LIPIcs.MFCS.2024.68.pdf
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  • 15 pages

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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
Keywords
  • hitting set
  • line-constrained
  • line-separable
  • unit disks
  • half-planes
  • coverage

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Abstract

Given a set P of n points and a set S of m disks in the plane, the disk hitting set problem asks for a smallest subset of P such that every disk of S contains at least one point in the subset. The problem is NP-hard. This paper considers a line-constrained version in which all disks have their centers on a line. We present an O(mlog²n+(n+m)log(n+m)) time algorithm for the problem. This improves the previous result of O(m²log m+(n+m)log(n+m)) time for the weighted case of the problem where every point of P has a weight and the objective is to minimize the total weight of the hitting set. Our algorithm also solves a more general line-separable problem with a single intersection property: The points of P and the disk centers are separated by a line 𝓁 and the boundary of every two disks intersect at most once on the side of 𝓁 containing P.

Cite As Get BibTex

Gang Liu and Haitao Wang. Unweighted Geometric Hitting Set for Line-Constrained Disks and Related Problems. In 49th International Symposium on Mathematical Foundations of Computer Science (MFCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 306, pp. 68:1-68:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.MFCS.2024.68

Author Details

Gang Liu
  • Kahlert School of Computing, University of Utah, Salt Lake City, UT, USA
Haitao Wang
  • Kahlert School of Computing, University of Utah, Salt Lake City, UT, USA

Funding

This research was supported in part by NSF under Grant CCF-2300356.

References

  1. Christoph Ambühl, Thomas Erlebach, Matús̆ Mihalák, and Marc Nunkesser. Constant-factor approximation for minimum-weight (connected) dominating sets in unit disk graphs. In Proceedings of the 9th International Conference on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), and the 10th International Conference on Randomization and Computation (RANDOM), pages 3-14, 2006. URL: https://doi.org/10.1007/11830924_3.
  2. Michael Ben-Or. Lower bounds for algebraic computation trees (preliminary report). In Proceedings of the 15th Annual ACM Symposium on Theory of Computing (STOC), pages 80-86, 1983. URL: https://doi.org/10.1145/800061.808735.
  3. Norbert Bus, Nabil H. Mustafa, and Saurabh Ray. Practical and efficient algorithms for the geometric hitting set problem. Discrete Applied Mathematics, 240:25-32, 2018. URL: https://doi.org/10.1016/j.dam.2017.12.018.
  4. Timothy M. Chan and Elyot Grant. Exact algorithms and APX-hardness results for geometric packing and covering problems. Computational Geometry: Theory and Applications, 47:112-124, 2014. URL: https://doi.org/10.1016/j.comgeo.2012.04.001.
  5. Timothy M. Chan and Dimitrios Skrepetos. All-pairs shortest paths in unit-disk graphs in slightly subquadratic time. In Proceedings of the 27th International Symposium on Algorithms and Computation (ISAAC), pages 24:1-24:13, 2016. URL: https://doi.org/10.4230/LIPIcs.ISAAC.2016.24.
  6. Timothy M. Chan and Da Wei Zheng. Hopcroft’s problem, log-star shaving, 2D fractional cascading, and decision trees. ACM Transactions on Algorithms, 2023. URL: https://doi.org/10.1145/3591357.
  7. Bernard Chazelle and Leonidas J. Guibas. Fractional cascading: I. A data structuring technique. Algorithmica, 1:133-162, 1986. URL: https://doi.org/10.1007/BF01840440.
  8. Francisco Claude, Gautam K. Das, Reza Dorrigiv, Stephane Durocher, Robert Fraser, Alejandro López-Ortiz, Bradford G. Nickerson, and Alejandro Salinger. An improved line-separable algorithm for discrete unit disk cover. Discrete Mathematics, Algorithms and Applications, 2:77-88, 2010. URL: https://doi.org/10.1142/S1793830910000486.
  9. Gruia Călinescu, Ion I. Măndoiu, Peng-Jun Wan, and Alexander Z. Zelikovsky. Selecting forwarding neighbors in wireless ad hoc networks. Mobile Networks and Applications, 9:101-111, 2004. URL: https://doi.org/10.1023/B:MONE.0000013622.63511.57.
  10. Gautam K. Das, Sandip Das, and Subhas C. Nandy. Homogeneous 2-hop broadcast in 2D. Computational Geometry: Theory and Applications, 43:182-190, 2010. URL: https://doi.org/10.1016/j.comgeo.2009.06.005.
  11. M. de Berg, O. Cheong, M. van Kreveld, and M. Overmars. Computational Geometry - Algorithms and Applications. Springer-Verlag, Berlin, 3rd edition, 2008. URL: https://doi.org/10.1007/978-3-540-77974-2.
  12. Stephane Durocher and Robert Fraser. Duality for geometric set cover and geometric hitting set problems on pseudodisks. In Proceedings of the 27th Canadian Conference on Computational Geometry (CCCG), 2015. URL: https://research.cs.queensu.ca/cccg2015/CCCG15-papers/10.pdf.
  13. Herbert Edelsbrunner, Leonidas J. Guibas, and J. Stolfi. Optimal point location in a monotone subdivision. SIAM Journal on Computing, 15(2):317-340, 1986. URL: https://doi.org/10.1137/0215023.
  14. Herbert Edelsbrunner and Ernst P. Mücke. Simulation of simplicity: A technique to cope with degenerate cases in geometric algorithms. ACM Transactions on Graphics, 9:66-104, 1990. URL: https://doi.org/10.1145/77635.77639.
  15. Guy Even, Dror Rawitz, and Shimon Shahar. Hitting sets when the VC-dimension is small. Information Processing Letters, 95:358-362, 2005. URL: https://doi.org/10.1016/j.ipl.2005.03.010.
  16. Shashidhara K. Ganjugunte. Geometric hitting sets and their variants. PhD thesis, Duke University, 2011. URL: https://dukespace.lib.duke.edu/server/api/core/bitstreams/0d37dabc-42a5-4bc8-b4e9-e69b263a10ca/content.
  17. Sariel Har-Peled and Mira Lee. Weighted geometric set cover problems revisited. Journal of Computational Geometry, 3:65-85, 2012. URL: https://doi.org/10.20382/jocg.v3i1a4.
  18. Richard M. Karp. Reducibility among combinatorial problems. Complexity of Computer Computations, pages 85-103, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  19. David G. Kirkpatrick. Efficient computation of continuous skeletons. In 20th Annual Symposium on Foundations of Computer Science (FOCS), pages 18-27, 1979. URL: https://doi.org/10.1109/SFCS.1979.15.
  20. David G. Kirkpatrick. Optimal search in planar subdivisions. SIAM Journal on Computing, 12(1):28-35, 1983. URL: https://doi.org/10.1137/0212002.
  21. Jian Li and Yifei Jin. A PTAS for the weighted unit disk cover problem. In Proceedings of the 42nd International Colloquium on Automata, Languages and Programming (ICALP), pages 898-909, 2015. URL: https://doi.org/10.1007/978-3-662-47672-7_73.
  22. Gang Liu and Haitao Wang. Geometric hitting set for line-constrained disks. In Proceedings of the 18th Algorithms and Data Structures Symposium (WADS), pages 574-587, 2023. URL: https://doi.org/10.1007/978-3-031-38906-1_38.
  23. Gang Liu and Haitao Wang. On the line-separable unit-disk coverage and related problems. In Proceedings of the 34th International Symposium on Algorithms and Computation (ISAAC), pages 51:1-51:14, 2023. Full version available at URL: https://arxiv.org/abs/2309.03162.
  24. El O. Mourad, Fohlin Helena, and Srivastav Anand. A randomised approximation algorithm for the hitting set problem. Theoretical Computer Science, 555:23-34, 2014. URL: https://doi.org/10.1007/978-3-642-36065-7_11.
  25. Nabil H. Mustafa and Saurabh Ray. Improved results on geometric hitting set problems. Discrete and Computational Geometry, 44:883-895, 2010. URL: https://doi.org/10.1007/s00454-010-9285-9.
  26. Logan Pedersen and Haitao Wang. Algorithms for the line-constrained disk coverage and related problems. Computational Geometry: Theory and Applications, 105-106:101883:1-18, 2022. URL: https://doi.org/10.1016/j.comgeo.2022.101883.
  27. Micha Sharir and Pankaj K. Agarwal. Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, New York, 1996. Google Scholar
  28. Haitao Wang and Jie Xue. Algorithms for halfplane coverage and related problems. In Proceedings of the 40th International Symposium on Computational Geometry (SoCG), pages 79:1-79:15, 2024. URL: https://doi.org/10.4230/LIPIcs.SoCG.2024.79.
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