Computing Partitions of Rectilinear Polygons with Minimum Stabbing Number

  • Stephane Durocher
  • Saeed Mehrabi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7434)


The stabbing number of a partition of a rectilinear polygon P into rectangles is the maximum number of rectangles stabbed by any axis-parallel line segment contained in P. We consider the problem of finding a rectangular partition with minimum stabbing number for a given rectilinear polygon P. First, we impose a conforming constraint on partitions: every vertex of every rectangle in the partition must lie on the polygon’s boundary. We show that finding a conforming rectangular partition of minimum stabbing number is NP-hard for rectilinear polygons with holes. We present a rounding method based on a linear programming relaxation resulting in a polynomial-time 2-approximation algorithm. We give an O(nlogn)-time algorithm to solve the problem exactly when P is a histogram (some edge in P can see every point in P) with n vertices. Next we relax the conforming constraint and show how to extend the first linear program to achieve a polynomial-time 2-approximation algorithm for the general problem, improving the approximation factor achieved by Abam, Aronov, de Berg, and Khosravi (ACM SoCG 2011).


Line Segment Vertical Edge Optimal Partition Linear Program Relaxation Initial Partition 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Abam, M.A., Aronov, B., de Berg, M., Khosravi, A.: Approximation algorithms for computing partitions with minimum stabbing number of rectilinear and simple polygons. In: Proc. ACM SoCG, pp. 407–416 (2011)Google Scholar
  2. 2.
    de Berg, M., Khosravi, A., Verdonschot, S., van der Weele, V.: On Rectilinear Partitions with Minimum Stabbing Number. In: Dehne, F., Iacono, J., Sack, J.-R. (eds.) WADS 2011. LNCS, vol. 6844, pp. 302–313. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    de Berg, M., van Kreveld, M.: Rectilinear decompositions with low stabbing number. Inf. Proc. Let. 52(4), 215–221 (1994)zbMATHCrossRefGoogle Scholar
  4. 4.
    Efrat, A., Erten, C., Kobourov, S.: Fixed-location circular arc drawing of planar graphs. J. Graph Alg. & Applications 11(1), 165–193 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Fekete, S., Lübbecke, M., Meijer, H.: Minimizing the stabbing number of matchings, trees, and triangulations. Disc. Comp. Geom. 40, 595–621 (2008)zbMATHCrossRefGoogle Scholar
  6. 6.
    Gourley, K., Green, D.: A polygon-to-rectangle conversion algorithm. IEEE Comp. Graphics & App. 3(1), 31–36 (1983)CrossRefGoogle Scholar
  7. 7.
    Hershberger, J., Suri, S.: A pedestrian approach to ray shooting: shoot a ray, take a walk. J. Alg. 18(3), 403–431 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Katz, M.J., Morgenstern, G.: Guarding orthogonal art galleries with sliding cameras. Inter. J. Comp. Geom. & App. 21(2), 241–250 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Knuth, D., Raghunathan, A.: The problem of compatible representatives. SIAM J. Disc. Math. 5(3), 422–427 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Lopez, M., Mehta, D.: Efficient decomposition of polygons into L-shapes with application to VLSI layouts. ACM Trans. Design Automation Elec. Sys. 1(3), 371–395 (1996)CrossRefGoogle Scholar
  11. 11.
    Punnen, A.: K-sum linear programming. J. Oper. Res. Soc. 43(4), 359–363 (1992)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Stephane Durocher
    • 1
  • Saeed Mehrabi
    • 1
  1. 1.Department of Computer ScienceUniversity of ManitobaWinnipegCanada

Personalised recommendations