Covering a Point Set by Two Disjoint Rectangles

  • Hee-Kap Ahn
  • Sang Won Bae
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5369)

Abstract

Given a set S of n points in the plane, the disjoint two-rectangle covering problem is to find a pair of disjoint rectangles such that their union contains S and the area of the larger rectangle is minimized. In this paper we consider two variants of this optimization problem: (1) the rectangles are free to rotate but must remain parallel to each other, and (2) one rectangle is axis-parallel but the other rectangle is allowed to have an arbitrary orientation. For both of the problems, we present O(n2logn)-time algorithms using O(n) space.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agarwal, P., Sharir, M.: Efficient algorithms for geometric optimization. ACM Comput. Surveys 30, 412–458 (1998)CrossRefGoogle Scholar
  2. 2.
    Bae, S.W., Lee, C., Ahn, H.-K., Choi, S., Chwa, K.-Y.: Maintaining extremal points and its applications to deciding optimal orientations. In: Proc. 18th Int. Sympos. Alg. Comput. (ISAAC), pp. 788–799 (2007)Google Scholar
  3. 3.
    Bespamyatnikh, S., Segal, M.: Covering a set of points by two axis-parallel boxes. Inform. Process. Lett. 75, 95–100 (2000)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Brodal, G.S., Jacob, R.: Dynamic planar convex hull. In: Proc. 43rd Annu. Found. Comput. Sci. (FOCS), pp. 617–626 (2002)Google Scholar
  5. 5.
    Frederickson, G., Johnson, D.: Generalized selection and ranking: sorted matrices. SIAM J. on Comput. 13, 14–30 (1984)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Hoffmann, F., Icking, C., Klein, R., Kriegel, K.: The polygon exploration problem. SIAM J. Comput. 31(2), 577–600 (2001)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Jaromczyk, J.W., Kowaluk, M.: Orientation indenpendent covering of point sets in R 2 with pairs of rectangles or optimal squares. In: Proc. 12th Euro. Workshop Comput. Geom. (EuroCG), pp. 54–61 (1996)Google Scholar
  8. 8.
    Kats, M., Kedem, K., Segal, M.: Discrete rectilinear 2-center problem. Comput. Geom.: Theory and Applications 15, 203–214 (2000)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Katz, M., Sharir, M.: An expander-based approach to geometric optimization. SIAM J. on Comput. 26, 1384–1408 (1997)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Megiddo, N.: Applying parallel computation algorithms in the design of serial algorithms. J. ACM 30, 852–865 (1983)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Saha, C., Das, S.: Covering a set of points in a plane using two parallel rectangles. In: Proc. 17th Int. Conf. Comput.: Theory and Appl (ICCTA), pp. 214–218 (2007)Google Scholar
  12. 12.
    Segal, M.: Covering point sets and accompanying problems. PhD thesis, Ben-Gurion University, Israel (1999)Google Scholar
  13. 13.
    Toussaint, G.: Solving geometric problems with the rotating calipers. In: Proc. IEEE MELECON 1983, Athens, Greece (1983)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Hee-Kap Ahn
    • 1
  • Sang Won Bae
    • 2
  1. 1.Department of Computer Science and EngineeringPOSTECHKorea
  2. 2.Division of Computer ScienceKAISTKorea

Personalised recommendations