ISAAC 2012: Algorithms and Computation pp 309-318

# Rectilinear Covering for Imprecise Input Points

(Extended Abstract)
• Hee-Kap Ahn
• Sang Won Bae
• Shin-ichi Tanigawa
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7676)

## Abstract

We consider the rectilinear k-center problem in the presence of impreciseness of input points. We assume that the input is a set S of n unit squares, possibly overlapping each other, each of which is interpreted as a measured point with an identical error bound under the L  ∞  metric on ℝ2. Our goal, in this work, is to analyze the worst situation with respect to the rectilinear k-center for a given set S of unit squares. For the purpose, we are interested in a value λ k (S) that is the minimum side length of k congruent squares by which any possible true point set from S can be covered. We show that, for k = 1 or 2, computing λ k (S) is equivalent to the problem of covering the input squares S completely by k squares, and thus one can solve the problem in linear time. However, for k ≥ 3, this is not the case, and we present an O(n logn)-time algorithm for computing λ 3(S). For structural observations, we introduce a new notion on geometric covering, namely the covering-family, which is of independent interest.

## Keywords

Side Length Voronoi Diagram Delaunay Triangulation Decision Algorithm Input Point
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.
Ahn, H.K., Knauer, C., Scherfenberg, M., Schlipf, L., Vigneron, A.: Computing the discrete Fréchet distance with imprecise input. Int. J. Comput. Geometry Appl. 22(1), 27–44 (2012)
2. 2.
Buchin, K., Löffler, M., Morin, P., Mulzer, W.: Preprocessing imprecise points for Delaunay triangulation: Simplified and extended. Algorithmica 61(3), 674–693 (2011)
3. 3.
Drezner, Z.: On the rectangular p-center problem. Naval Res. Logist. 34(2), 229–234 (1987)
4. 4.
Frederickson, G., Johnson, D.: Generalized selection and ranking: Sorted matrices. SIAM J. Comput. 13(1), 14–30 (1984)
5. 5.
Hoffmann, M.: Covering polygons with few rectangles. In: Proc. 17th Euro. Workshop Comp. Geom. (EuroCG 2001), pp. 39–42 (2001)Google Scholar
6. 6.
Hoffmann, M.: A simple linear algorithm for computing rectilinear 3-centers. Comput. Geom. Theory Appl. 31, 150–165 (2005)
7. 7.
Jadhav, S., Mukhopadhyay, A., Bhattacharya, B.K.: An optimal algorithm for the intersection radius of a set of convex polygons. J. Algo. 20, 244–267 (1996)
8. 8.
Khanban, A.A., Edalat, A.: Computing Delaunay triangulation with imprecise input data. In: Proc. 15th Canadian Conf. Comput. Geom., pp. 94–97 (2003)Google Scholar
9. 9.
Knauer, C., Löffler, M., Scherfenberg, M., Wolle, T.: The directed Hausdorff distance between imprecise point sets. Theoretical Comput. Sci. 412(32), 4173–4186 (2011)
10. 10.
Ko, M., Lee, R., Chang, J.: An optimal approximation algorithm for the rectilinear m-center problem. Algorithmica 5, 341–352 (1990)
11. 11.
Ko, M., Lee, R., Chang, J.: Rectilinear m-center problem. Naval Res. Logist. 37(3), 419–427 (1990)
12. 12.
Löffler, M.: Data Imprecision in Computational Geometry. Ph.D. thesis, Utrecht University (2009)Google Scholar
13. 13.
Löffler, M., Snoeyink, J.: Delaunay triangulation of imprecise points in linear time after preprocessing. Comput. Geom.: Theory and Appl. 43(3), 234–242 (2010)
14. 14.
Löffler, M., van Kreveld, M.J.: Largest and smallest tours and convex hulls for imprecise points. In: Proc. 10th Scandinavian Workshop Algo. Theory, pp. 375–387 (2006)Google Scholar
15. 15.
Megiddo, M., Supowit, K.J.: On the complexity of some common geometric location problems. SIAM J. Comput. 13(1), 182–196 (1984)
16. 16.
Segal, M.: On piercing sets of axis-parallel rectangles and rings. Int. J. Comput. Geometry Appl. 9(3), 219–234 (1999)
17. 17.
Sember, J., Evans, W.: Guaranteed Voronoi diagrams of uncertain sites. In: Proc. 20th Canadian Conf. Comput. (2008)Google Scholar
18. 18.
Sharir, M., Welzl, E.: Rectilinear and polygonal p-piercing and p-center problems. In: Proc. 12th Annu. Sympos. Comp. Geom (SoCG 1996), pp. 122–132 (1996)Google Scholar

## Authors and Affiliations

• Hee-Kap Ahn
• 1
• Sang Won Bae
• 2
• Shin-ichi Tanigawa
• 3
1. 1.Department of Computer Science and EngineeringPOSTECHPohangKorea
2. 2.Department of Computer ScienceKyonggi UniversitySuwonKorea
3. 3.Research Institute for Mathematical ScienceKyoto UniversityKyotoJapan