Approximation algorithms for covering polygons with squares and similar problems

Extended abstract
  • Christos Levcopoulos
  • Joachim Gudmundsson
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1269)


We consider the problem of covering arbitrary polygons, without any acute interior angles, using a preferably minimum number of squares. The squares must lie entirely within the polygon. Let P be an arbitrary input polygon, with n vertices, coverable by squares. Let μ(P) denote the minimum number of squares required to cover P. In the first part of this paper we present an algorithm which guarantees a constant (14) approximation factor running in O(n2(P)) time. As a corollary we obtain the first polynomial-time, constant-factor approximation algorithm for “fat” rectangular coverings. In the second part we show an O(n log n+μ(P)) time algorithm which produces at most 11n+μ(P) squares to cover P. In the hole-free case this algorithm runs in linear time and produces a cover which is within an O(α(n)) approximation factor of the optimal, where α(n) is the extremely slowly growing inverse of Ackermann's function. In parallel our algorithm runs in O(log n) randomized time using O(max(μ(P), n)) processors.


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  1. 1.
    L.J. Aupperle, H.E. Conn, J.M. Keil and J. O'Rourke, Covering Orthogonal Polygons with Squares, 26th Annual Allerton Conference on Communication, Control and Computation, 1988.Google Scholar
  2. 2.
    R. Bar-Yehuda and E. Ben-Hanoch, A Linear-Time Algorithm for Covering Simple Polygons with Similar rectangles, International Journal of Computational Geometry & Applications, vol. 6, no 1, 1996.Google Scholar
  3. 3.
    B.M. Chazelle, Computational Geometry and Convexity, Ph.D. Thesis, Dept. Comp. Sci., Yale University, New Haven, CT, 1979. Carnegie-Mellon Univ. Report CS-80-150.Google Scholar
  4. 4.
    J. Snoeyink, C.A. Wang and F. Chin, Finding the Medial Axis of a Simple Polygon in Linear-Time, ISAAC '95, Cairns, Australia, 1995 (LNCS 1006 Springer-Verlag).Google Scholar
  5. 5.
    C. Levcopoulos and J. Gudmundsson, Close Approximation of Minimum Rectangular Coverings, FST&TCS-16, Hyderabad, India, 1996 (LNCS, Springer-Verlag).Google Scholar
  6. 6.
    C. Levcopoulos and J. Gudmundsson, Approximation Algorithms for Covering Polygons with Squares and Similar Problems, LU-CS-TR:96-181, Dept. of Comp. Sci., Lund University, 1996.Google Scholar
  7. 7.
    A. Hegedüs, Algorithms for covering polygons by rectangles, Computer Aided Design, vol. 14, no 5, 1982.Google Scholar
  8. 8.
    S. Hart and M. Sharir, Nonlinearity of Davenport-Schinzel Sequences and of Generalized Path Compression Schemes, Tech. Report 84-011, The Eskenasy Institute of Comp. Sci., Tel Aviv University, August 1984.Google Scholar
  9. 9.
    D.G. Kirkpatrick, Efficient computation of continuous skeletons, 20th Annual IEEE Symposium on Foundation of Computer Science, 1979.Google Scholar
  10. 10.
    J.M. Keil and J.-R. Sack, Minimum Decompositions of Polygonal Objects, Machine Intelligence and Pattern Recognition vol. 2: Computational Geometry, pp. 197–215, Elsevier Science Publishers B.V., 1985.Google Scholar
  11. 11.
    C. Levcopoulos, A Fast Heuristic for Covering Polygons by Rectangles, FCT'85, Cottbus, GDR, 1985 (LNCS 199, Springer-Verlag).Google Scholar
  12. 12.
    C. Levcopoulos, Improved Bounds for Covering General Polygons with Rectangles, FST&TCS-7, Pune, India, 1987 (LNCS 287, Springer-Verlag).Google Scholar
  13. 13.
    D. Morita, Finding a Minimal Cover for Binary Images: an Optimal Parallel Algorithm, Tech. Report No. 88-946, Dept. of Comp. Sci., Cornell University, 1988.Google Scholar
  14. 14.
    J. O'Rourke and K.J. Supowit, Some NP-hard Polygon Decomposition Problems, IEEE Transactions on Information Theory, vol. IT-29, pp. 181–190, 1983.Google Scholar
  15. 15.
    F.P. Preparata and M.I. Shamos, Computational Geometry, New York, Springer-Verlag, 1985.Google Scholar
  16. 16.
    S. Rajasekaran and S. Ramaswami, Optimal Parallel Randomized Algorithms for the Voronoi Diagram of Line Segments in the Plane and Related Problems, In Proc. ACM Symposium on Computational Geometry, Stony Brook, New York, 1994.Google Scholar
  17. 17.
    D.S. Scott and S.S. Iyengar, TID: a Translation Invariant Data Structure for Storing Images, Comm. of the ACM, vol. 29, no. 5, 1986.Google Scholar
  18. 18.
    A. Wiernik, Planar Realization of Nonlinear Davenport-Schinzel Sequences by Segments, 27th IEEE Symposium on Computer Science, 1986.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Christos Levcopoulos
    • 1
  • Joachim Gudmundsson
    • 1
  1. 1.Department of Computer ScienceLund UniversityLundSweden

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