FAW 2009: Frontiers in Algorithmics pp 132-140

# Square and Rectangle Covering with Outliers

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5598)

## 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 p log p 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 + p log p − 1 k) time for p = 2,3. In all cases, our algorithms use O(n) space.

## Keywords

Extreme Point Covering Problem Covering Algorithm Optimal Index Vertical Slab
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. 1.
Aggarwal, A., Imai, H., Katoh, N., Suri, S.: Finding k points with minimum diameter and related problems. J. Algorithms 12, 38–56 (1991)
2. 2.
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)Google Scholar
3. 3.
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)Google Scholar
4. 4.
Bespamyatnikh, S., Segal, M.: Covering a set of points by two axis–parallel boxes. Inform. Proc. Lett, 95–100 (2000)Google Scholar
5. 5.
Chan, T.M.: Geometric applications of a randomized optimization technique. Discrete Comput. Geom. 22(4), 547–567 (1999)
6. 6.
Chazelle, B.: An algorithm for segment-dragging and its implementation. Algorithmica 3, 205–221 (1988)
7. 7.
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press, Cambridge (2001)
8. 8.
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)
9. 9.
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)Google Scholar
10. 10.
Katz, M.J., Kedem, K., Segal, M.: Discrete rectilinear 2-center problems. Comput. Geom. Theory Appl. 15, 203–214 (2000)
11. 11.
Matoušek, J., Welzl, E., Sharir, M.: A subexponential bound for linear programming and related problems. Algorithmica 16, 365–384 (1996)
12. 12.
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)Google Scholar
13. 13.
Segal, M.: Lower bounds for covering problems. Journal of Mathematical Modelling and Algorithms 1, 17–29 (2002)
14. 14.
Segal, M., Kedem, K.: Enclosing k points in the smallest axis parallel rectangle. Inform. Process. Lett. 65, 95–99 (1998)
15. 15.
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)Google Scholar