Abstract
Given a set \({\cal P}\) of n points and a set \({\cal D}\) of m unit disks on a 2-dimensional plane, the discrete unit disk cover (DUDC) problem is (i) to check whether each point in \({\cal P}\) is covered by at least one disk in \({\cal D}\) or not and (ii) if so, then find a minimum cardinality subset \({\cal D}^* \subseteq {\cal D}\) such that unit disks in \({\cal D}^*\) cover all the points in \({\cal P}\). The discrete unit disk cover problem is a geometric version of the general set cover problem which is NP-hard [14]. The general set cover problem is not approximable within \(c \log |{\cal P}|\), for some constant c, but the DUDC problem was shown to admit a constant factor approximation. In this paper, we provide an algorithm with constant approximation factor 18. The running time of the proposed algorithm is O(n logn + m logm + mn). The previous best known tractable solution for the same problem was a 22-factor approximation algorithm with running time O(m 2 n 4).
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References
Agarwal, P.K., Procopiuc, C.M.: Exact and approximation algorithms for clustering. Algorithmica 33, 201–226 (2002)
Agarwal, P.K., Sharir, M.: Efficient algorithms for geometric optimization. ACM Computing Serveys 30, 412–458 (1998)
de Berg, M., Kreveld, M.V., Overmars, M., Schwarzkopf, O.: Computational Geometry Algorithms and Applications. Springer, Heidelberg (1997)
Brönnimann, H., Goodrich, M.: Almost optimal set covers in finite VC-dimension. Disc. and Comp. Geom. 14(1), 463–479 (1995)
Claude, F., Das, G.K., Dorrigiv, R., Durocher, S., Fraser, R., Lopez-Ortiz, A., Nickerson, B.G., Salinger, A.: An improved line-separable algorithm for discrete unit disk cover. Disc. Math., Alg. and Appl. 2, 77–87 (2010)
Călinescu, G., Măndoiu, I., Wan, P.J., Zelikovsky, A.: Selecting forwarding neighbours in wireless ad hoc networks. Mobile Networks and Applications 9, 101–111 (2004)
Carmi, P., Katz, M.J., Lev-Tov, N.: Covering points by unit disks of fixed location. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 644–655. Springer, Heidelberg (2007)
Chvátal, V.: A Greedy Heuristic for the Set-Covering Problem. Mathematics of Operations Research 4(3), 233–235 (1979)
Fowler, R., Paterson, M., Tanimoto, S.: Optimal packing and covering in the plane are NP-complete. Information Processing Letters 12, 133–137 (1981)
Frederickson, G.: Fast algorithms for shortest paths in planar graphs, with applications. SIAM J. on Computing 16, 1004–1022 (1987)
Gonzalez, T.: Covering a set of points in multidimensional space. Information Processing Letters 40, 181–188 (1991)
Hochbaum, D., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32, 130–136 (1985)
Hwang, R., Lee, R., Chang, R.: The generalized searching over separators strategy to solve some NP-hard problems in subexponential time. Algorithmica 9, 398–423 (1993)
Johnson, D.S.: The NP-completeness column:An ongoing guide. J. Algorithms 3, 182–195 (1982)
Koutis, I., Miller, G.L.: A linear work, o(n 1/6) time, parallel algorithm for solving planar Laplacians. In: Proc. of the 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1002–1011 (2007)
Mustafa, N.H., Ray, S.: Improved results on geometric hitting set problems. Discrete and Computational Geometry 44(4), 883–895 (2010)
Narayanappa, S., Vojtechovsky, P.: An improved approximation factor for the unit disk covering problem. In: Proc. of the 18th Canadian Conference on Computational Geometry, pp. 15–18 (2006)
Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proc. of the 29th ACM Symposium on Theory of Computing, pp. 475–484 (1997)
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Das, G.K., Fraser, R., Lòpez-Ortiz, A., Nickerson, B.G. (2011). On the Discrete Unit Disk Cover Problem. In: Katoh, N., Kumar, A. (eds) WALCOM: Algorithms and Computation. WALCOM 2011. Lecture Notes in Computer Science, vol 6552. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19094-0_16
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DOI: https://doi.org/10.1007/978-3-642-19094-0_16
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