Abstract
An NP-hard problem is considered to stab a given set of n straight line segments on the plane with the smallest size subset of disks of fixed radii \(r>0,\) where the set of segments forms a straight line drawing \(G=(V,E)\) of a planar graph without proper edge crossings. To the best of our knowledge, only 100-approximation \(O(n^4\log n)\)-time algorithm is known (Kobylkin, 2018) for this problem. Moreover, when segments of E are axis-parallel, 8-approximation is proposed (Dash et al., 2012), working in \(O(n\log n)\) time. In this work another special setting is considered of the problem where G belongs to classes of special plane graphs, which are of interest in network applications. Namely, three fast \(O(n^{3/2}\log ^2n)\)-expected time algorithms are proposed: a 10-approximate algorithm for the problem, considered on edge sets of minimum Euclidean spanning trees, a 12-approximate algorithm for edge sets of relative neighborhood graphs and 14-approximate algorithm for edge sets of Gabriel graphs. The paper extends recent work (Kobylkin et al. 2019) where \(O(n^2)\)-time approximation algorithms are proposed with the same constant approximation factors for the problem on those three classes of sets of segments.
The work is carried out within the research, conducted at the Ural Mathematical Center. It is also supported by the Russian Foundation for Basic Research, project №-19-07-01243. The paper is a substantially extended version of the short paper Kobylkin, K., Dryakhlova, I.: Practical approximation algorithms for stabbing special families of line segments with equal disks. In: Kotsireas, I., Pardalos, P. (eds.) Learning and Intelligent Optimization, LION 2020. Lecture Notes in Computer Science, vol. 12096. Springer, Cham.
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Notes
- 1.
Its C++ implementation is built in the CGAL library (see https://www.cgal.org/), providing robust geometric computations.
References
Boissonnat, J.D., Wormser, C., Yvinec, M.: Curved Voronoi diagrams. In: Boissonnat, J.D., Teillaud, M. (eds.) Effective Computational Geometry for Curves and Surfaces, pp. 67–116. Springer, Heidelberg (2006). https://doi.org/10.1007/978-3-540-33259-6_2
Chan, T.M.: Dynamic geometric data structures via shallow cuttings. In: Proceedings of the 35th International Symposium on Computational Geometry, SoCG 2019, pp. 24:1–24:13 (2019)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press, Cambridge (2009)
Dash, D., Bishnu, A., Gupta, A., Nandy, S.: Approximation algorithms for deployment of sensors for line segment coverage in wireless sensor networks. Wirel. Netw. 19(5), 857–870 (2012)
Devillers, O.: The Delaunay hierarchy. Int. J. Found. Comput. Sci. 13, 163–180 (2002)
Jaromczyk, J., Toussaint, G.: Relative neighborhood graphs and their relatives. Proc. IEEE 80(9), 1502–1517 (1992)
Karavelas, M.I.: A robust and efficient implementation for the segment Voronoi diagram. In: Proceedings of the 1st International Symposium on Voronoi Diagrams in Science and Engineering, Tokyo, pp. 51–62 (2004)
Karavelas, M., Yvinec, M.: The Voronoi Diagram of Convex Objects in the Plane. Research report. RR-5023, INRIA. 27 p. (2003)
Klein, R., Mehlhorn, K., Meiser, S.: Randomized incremental construction of abstract Voronoi diagrams. Comput. Geom. 3(3), 157–184 (1993)
Kobylkin, K.: Stabbing line segments with disks: complexity and approximation algorithms. In: van der Aalst, W.M.P., et al. (eds.) AIST 2017. LNCS, vol. 10716, pp. 356–367. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-73013-4_33
Kobylkin, K.: Constant factor approximation for intersecting line segments with disks. In: Battiti, R., Brunato, M., Kotsireas, I., Pardalos, P.M. (eds.) LION 12 2018. LNCS, vol. 11353, pp. 447–454. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-05348-2_39
Kobylkin, K.: Efficient constant factor approximation algorithms for stabbing line segments with equal disks, CoRR abs/1803.08341, 31 p. (2018). https://arxiv.org/pdf/1803.08341.pdf
Kobylkin, K., Dryakhlova, I.: Approximation algorithms for piercing special families of hippodromes: an extended abstract. In: Khachay, M., Kochetov, Y., Pardalos, P. (eds.) MOTOR 2019. LNCS, vol. 11548, pp. 565–580. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-22629-9_40
Matula, D., Sokal, R.: Properties of Gabriel graphs relevant to geographic variation research and the clustering of points in the plane. Geogr. Anal. 12(3), 205–222 (1980)
Motwani, R., Raghavan, P.: Randomized Algorithms, 476 p. Cambridge University Press, Cambridge (1995)
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Kobylkin, K. (2021). Fast Approximation Algorithms for Stabbing Special Families of Line Segments with Equal Disks. In: van der Aalst, W.M.P., et al. Analysis of Images, Social Networks and Texts. AIST 2020. Lecture Notes in Computer Science(), vol 12602. Springer, Cham. https://doi.org/10.1007/978-3-030-72610-2_32
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