Abstract
For a set of n points in the plane, we consider the axis–aligned (p,k)-Box Covering problem: Find p axis-aligned, pairwise disjoint boxes that together contain exactly n − k points. Here, our boxes are either squares or rectangles, and we want to minimize the area of the largest box. For squares, we present algorithms that find the solution in O(n + klogk) time for p = 1, and in O(nlogn + k plogp k) time for p = 2,3. For rectangles we have running times of O(n + k 3) for p = 1 and O(nlogn + k 2 + plogp − 1 k) time for p = 2,3. In all cases, our algorithms use O(n) space.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Aggarwal, A., Imai, H., Katoh, N., Suri, S.: Finding k points with minimum diameter and related problems. J. Algorithms 12, 38–56 (1991)
Ahn, H.-K., Bae, S.W.: Covering a point set by two disjoint rectangles. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 728–739. Springer, Heidelberg (2008)
Atanassov, R., Bose, P., Couture, M., Maheshwari, A., Morin, P., Paquette, M., Smid, M., Wuhrer, S.: Algorithms for optimal outlier removal. J. Discrete Alg. (to appear)
Bespamyatnikh, S., Segal, M.: Covering a set of points by two axis–parallel boxes. Inform. Proc. Lett, 95–100 (2000)
Chan, T.M.: Geometric applications of a randomized optimization technique. Discrete Comput. Geom. 22(4), 547–567 (1999)
Chazelle, B.: An algorithm for segment-dragging and its implementation. Algorithmica 3, 205–221 (1988)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)
Das, S., Goswamib, P.P., Nandy, S.C.: Smallest k-point enclosing rectangle and square of arbitrary orientation. Inform. Proc. Lett. 94(6), 259–266 (2005)
Jaromczyk, J.W., Kowaluk, M.: Orientation independent covering of point sets in R 2 with pairs of rectangles or optimal squares. In: Abstracts 12th European Workshop Comput. Geom., pp. 77–84. Universität Münster (1996)
Katz, M.J., Kedem, K., Segal, M.: Discrete rectilinear 2-center problems. Comput. Geom. Theory Appl. 15, 203–214 (2000)
Matoušek, J., Welzl, E., Sharir, M.: A subexponential bound for linear programming and related problems. Algorithmica 16, 365–384 (1996)
Saha, C., Das, S.: Covering a set of points in a plane using two parallel rectangles. In: ICCTA 2007: Proceedings of the International Conference on Computing: Theory and Applications, pp. 214–218 (2007)
Segal, M.: Lower bounds for covering problems. Journal of Mathematical Modelling and Algorithms 1, 17–29 (2002)
Segal, M., Kedem, K.: Enclosing k points in the smallest axis parallel rectangle. Inform. Process. Lett. 65, 95–99 (1998)
Sharir, M., Welzl, E.: Rectilinear and polygonal p-piercing and p-center problems. In: Proc. 12th Annu. ACM Sympos. Comput. Geom, pp. 122–132 (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Ahn, HK., Bae, S.W., Kim, SS., Korman, M., Reinbacher, I., Son, W. (2009). Square and Rectangle Covering with Outliers . In: Deng, X., Hopcroft, J.E., Xue, J. (eds) Frontiers in Algorithmics. FAW 2009. Lecture Notes in Computer Science, vol 5598. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02270-8_15
Download citation
DOI: https://doi.org/10.1007/978-3-642-02270-8_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02269-2
Online ISBN: 978-3-642-02270-8
eBook Packages: Computer ScienceComputer Science (R0)