Skip to main content
SpringerLink
Log in

A Scheme for Computing Minimum Covers within Simple Regions

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

Let X be a simple region (e.g., a simple polygon), and let Q be a set of points in X. Let O be a convex object, such as a disk, a square, or an equilateral triangle. We present a scheme for computing a minimum cover of Q, consisting of homothets of O contained in X. In particular, a minimum disk cover of Q with respect to X, can be computed in polynomial time.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Alt, H., Arkin, E.M., Brönnimann, H., Erickson, J., Fekete, S.P., Knauer, C., Lenchner, J., Mitchell, J.S.B., Whittlesey, K.: Minimum-cost coverage of point sets by disks. In: Proc. 22nd Sympos. Comput. Geom., pp. 449–458 (2006)

    Google Scholar 

  2. Brönnimann, H., Goodrich, M.T.: Almost optimal set covers in finite VC-dimension. Discrete Comput. Geom. 14, 463–479 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Calinescu, G., Mandoiu, I.I., Wan, P.-J., Zelikovsky, A.: Selecting forwarding neighbors in wireless ad hoc networks. Monet 9(2), 101–111 (2004)

    Google Scholar 

  4. Carmi, P., Katz, M.J., Lev-Tov, N.: Covering points by unit disks of fixed location. In: Proc. 18th Internat. Sympos. on Algorithms and Computation, pp. 644–655 (2007)

    Google Scholar 

  5. Chew, L.P., Drysdale, R.L.: Voronoi diagrams based on convex distance functions. In: Proc. 1st Sympos. Comput. Geom., pp. 235–244 (1985)

    Google Scholar 

  6. Chin, F., Snoeyink, J., Wang, C.A.: Finding the medial axis of a simple polygon in linear time. Discrete Comput. Geom. 21(3), 405–420 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Gavril, F.: Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph. SIAM J. Comput. 1, 180–187 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  8. Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs, 1st edn. Academic Press, New York (1980); 2nd edn.: Annals of Discrete Mathematics, vol. 57. Elsevier (2004)

    MATH  Google Scholar 

  9. Grötschel, M., Lovász, L., Schrijver, A.: Polynomial algorithms for Perfect Graphs. In: Berge, C., Chvátal, V. (eds.) Topics on Perfect Graphs. Annals of Discrete Mathematics, vol. 21, pp. 325–356. North-Holland, Amsterdam (1984)

    Chapter  Google Scholar 

  10. Halperin, D., Linhart, C.: The minimum enclosing disk with obstacles. Manuscript (1999)

  11. Halperin, D., Sharir, M., Goldberg, K.: The 2-center problem with obstacles. J. Algorithms 42, 109–134 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hochbaum, D.S., Maas, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32, 130–136 (1985)

    Article  MATH  Google Scholar 

  13. Hurtado, F., Sacristán, V., Toussaint, G.: Some constrained minimax and maximin location problems. Stud. Locat. Anal. 15, 17–35 (2000)

    MATH  Google Scholar 

  14. Hwang, F.K.: An O(nlog n) algorithm for rectilinear minimal spanning trees. J. ACM 26(2), 177–182 (1979)

    Article  MATH  Google Scholar 

  15. Katz, M.J., Morgenstern, G.: Guarding orthogonal art galleries with sliding cameras. In: Proc. 25th European Workshop on Comput. Geom., pp. 159–162 (2009)

    Google Scholar 

  16. Katz, M.J., Roisman, G.S.: On guarding the vertices of rectilinear domains. Comput. Geom. Theory Appl. 39(3), 219–228 (2008)

    MATH  MathSciNet  Google Scholar 

  17. Lee, D.T.: Two-dimensional Voronoi diagrams in the L p -metric. J. ACM 27(4), 604–618 (1980)

    Article  MATH  Google Scholar 

  18. Lee, D.T., Wong, C.K.: Voronoi diagrams in L 1 (L ∞) Metrics with 2-dimensional storage applications. SIAM J. Comput. 9(1), 200–211 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  19. Motwani, R., Raghunathan, A., Saran, H.: Covering orthogonal polygons with star polygons: the perfect graph approach. J. Comput. Syst. Sci. 40, 19–48 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  20. Motwani, R., Raghunathan, A., Saran, H.: Perfect graphs and orthogonally convex covers. SIAM J. Discrete Math. 2(3), 371–392 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  21. Mustafa, N.H., Ray, P.S.: PTAS for geometric hitting set problems via local search. In: Proc. 25th Sympos. Comput. Geom., pp. 17–22 (2009)

    Google Scholar 

  22. Narayanappa, S., Vojtechovsky, P.: An improved approximation factor for the unit disk covering problem. In: Proc. 18th Canadian Conf. Comput. Geom., pp. 15–18 (2006)

    Google Scholar 

  23. Worman, C., Keil, J.M.: Polygon decomposition and the orthogonal art gallery problem. Int. J. Comput. Geom. Appl. 17(2), 105–138 (2007)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Ben-Gurion University, Beer-Sheva, 84105, Israel

    Matthew J. Katz & Gila Morgenstern

Authors
  1. Matthew J. Katz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gila Morgenstern
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Matthew J. Katz.

Additional information

A preliminary version of this paper has appeared in the 11th Algorithms and Data Structures Symposium, WADS 2009.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Katz, M.J., Morgenstern, G. A Scheme for Computing Minimum Covers within Simple Regions. Algorithmica 62, 349–360 (2012). https://doi.org/10.1007/s00453-010-9458-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-010-9458-1

Keywords

Search

Navigation