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# Geometric Cover with Outliers Removal

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LIPIcs.STACS.2021.39.pdf
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## Cite As

Zhengyang Guo and Yi Li. Geometric Cover with Outliers Removal. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 39:1-39:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)
https://doi.org/10.4230/LIPIcs.STACS.2021.39

## Abstract

We study the problem of partial geometric cover, which asks to find the minimum number of geometric objects (unit squares and unit disks in this work) that cover at least (n-t) of n given planar points, where 0 ≤ t ≤ n/2. When t = 0, the problem is the classical geometric cover problem, for which many existing works adopt a general framework called the shifting strategy. The shifting strategy is a divide and conquer paradigm which partitions the plane into equal-width strips, applies a local algorithm on each strip and then merges the local solutions with only a small loss on the overall approximation ratio. A challenge to extend the shifting strategy to the case of outliers is to determine the number of outliers in each strip. We develop a shifting strategy incorporating the outlier distribution, which runs in O(tn log n) time. We also develop local algorithms on strips for the outliers case, improving the running time over previous algorithms, and consequently obtain approximation algorithms to the partial geometric cover.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Design and analysis of algorithms
##### Keywords
• Geometric Cover
• Unit Square Cover
• Unit Disk Cover
• Shifting Strategy
• Outliers Detection
• Computational Geometry

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## References

1. Christoph Ambühl, Thomas Erlebach, Matúš Mihalák, and Marc Nunkesser. Constant-factor approximation for minimum-weight (connected) dominating sets in unit disk graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 3-14. Springer, 2006.
2. Rossen Atanassov, Prosenjit Bose, Mathieu Couture, Anil Maheshwari, Pat Morin, Michel Paquette, Michiel Smid, and Stefanie Wuhrer. Algorithms for optimal outlier removal. Journal of discrete algorithms, 7(2):239-248, 2009.
3. Ahmad Biniaz, Paul Liu, Anil Maheshwari, and Michiel Smid. Approximation algorithms for the unit disk cover problem in 2D and 3D. Computational Geometry, 60:8-18, 2017.
4. Moses Charikar, Chandra Chekuri, Tomás Feder, and Rajeev Motwani. Incremental clustering and dynamic information retrieval. SIAM Journal on Computing, 33(6):1417-1440, 2004.
5. Adrian Dumitrescu, Anirban Ghosh, and Csaba D Tóth. Online unit covering in euclidean space. Theoretical Computer Science, 809:218-230, 2020.
6. David Eppstein and Jeff Erickson. Iterated nearest neighbors and finding minimal polytopes. Discrete & Computational Geometry, 11(3):321-350, 1994.
7. Robert J Fowler, Michael S Paterson, and Steven L Tanimoto. Optimal packing and covering in the plane are NP-complete. Information processing letters, 12(3):133-137, 1981.
8. Massimo Franceschetti, Matthew Cook, and Jehoshua Bruck. A geometric theorem for approximate disk covering algorithms. Technical Report ETR035, California Institute of Technology, 2001.
9. Robert Fraser and Alejandro López-Ortiz. The within-strip discrete unit disk cover problem. Theoretical Computer Science, 674:99-115, 2017.
10. Bin Fu, Zhixiang Chen, and Mahdi Abdelguerfi. An almost linear time 2.8334-approximation algorithm for the disc covering problem. In International Conference on Algorithmic Applications in Management, pages 317-326. Springer, 2007.
11. Rajiv Gandhi, Samir Khuller, and Aravind Srinivasan. Approximation algorithms for partial covering problems. Journal of Algorithms, 53(1):55-84, 2004.
12. Hossein Ghasemalizadeh and Mohammadreza Razzazi. An improved approximation algorithm for the most points covering problem. Theory of Computing Systems, 50(3):545-558, 2012.
13. Anirban Ghosh, Brian Hicks, and Ronald Shevchenko. Unit disk cover for massive point sets. In International Symposium on Experimental Algorithms, pages 142-157. Springer, 2019.
14. Teofilo F Gonzalez. Covering a set of points in multidimensional space. Information processing letters, 40(4):181-188, 1991.
15. Sudipto Guha, Yi Li, and Qin Zhang. Distributed partial clustering. ACM Transactions on Parallel Computing (TOPC), 6(3):1-20, 2019.
16. Sariel Har-Peled. On complexity, sampling, and ε-nets and ε-samples. Approximation Algorithm in Geometry, 2010.
17. Alice Héliou, Martine Léonard, Laurent Mouchard, and Mikael Salson. Efficient dynamic range minimum query. Theoretical Computer Science, 656:108-117, 2016.
18. Dorit S Hochbaum and Wolfgang Maass. Approximation schemes for covering and packing problems in image processing and VLSI. Journal of the ACM (JACM), 32(1):130-136, 1985.
19. Tanmay Inamdar. Local search for geometric partial covering problems. In Proceedings of the 31st Canadian Conference on Computational Geometry, pages 242-249, 2019.
20. Paul Liu and Daniel Lu. A fast 25/6-approximation for the minimum unit disk cover problem, 2014. URL: http://arxiv.org/abs/1406.3838.
21. Nina Mishra, Rajeev Motwani, and Sergei Vassilvitskii. Sublinear projective clustering with outliers. In 15th Annual Fall Workshop on Computational Geometry and Visualization, page 45. Citeseer, 2005.
22. Sada Narayanappa and Petr Vojtechovsky. An improved approximation factor for the unit disk covering problem. In Proceedings of the 18th Canadian Conference on Computational Geometry, pages 15-18, 2006.
23. Kaveh Pahlavan and Allen H Levesque. Wireless data communications. Proceedings of the IEEE, 82(9):1398-1430, 1994.
24. Michael Segal and Klara Kedem. Enclosing k points in the smallest axis parallel rectangle. Information Processing Letters, 65(2):95-99, 1998.
25. Steven L Tanimoto and Robert J Fowler. Covering image subsets with patches. In Proceedings of the fifty-first International Conference on Pattern Recognition, pages 835-839, 1980.
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