Discrete & Computational Geometry

, Volume 45, Issue 4, pp 617–646 | Cite as

Binary Plane Partitions for Disjoint Line Segments



A binary space partition (BSP) for a set of disjoint objects in Euclidean space is a recursive decomposition where each step partitions the space (and possibly some of the objects) along a hyperplane and recurses on the objects clipped in each of the two open half-spaces. The size of a BSP is defined as the number of resulting fragments of the input objects. It is shown that every set of n disjoint line segments in the plane admits a BSP of size O(nlog n/log log n). This bound is the best possible.


Binary space partitions Line segments 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agarwal, P.K., Grove, E.F., Murali, T.M., Vitter, J.S.: Binary space partitions for fat rectangles. SIAM J. Comput. 29, 1422–1448 (2000) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Ar, S., Chazelle, B., Tal, A.: Self-customized BSP trees for collision detection. Comput. Geom. Theory Appl. 15(1–3), 91–102 (2000) Google Scholar
  3. 3.
    Ar, S., Montag, G., Tal, A.: Deferred, self-organizing BSP trees. Comput. Graph. Forum 21(3), 269–278 (2002) CrossRefGoogle Scholar
  4. 4.
    Arya, S.: Binary space partitions for axis-parallel line segments: size-height tradeoffs. Inf. Process. Lett. 84(4), 201–206 (2002) CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Asano, T., de Berg, M., Cheong, O., Guibas, L.J., Snoeyink, J., Tamaki, H.: Spanning trees crossing few barriers. Discrete Comput. Geom. 30, 591–606 (2003) MATHMathSciNetGoogle Scholar
  6. 6.
    de Berg, M.: Linear size binary space partitions for uncluttered scenes. Algorithmica 28(3), 353–366 (2000) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    de Berg, M., Streppel, M.: Approximate range searching using binary space partitions. Comput. Geom. Theory Appl. 33(3), 139–151 (2006) MATHGoogle Scholar
  8. 8.
    de Berg, M., de Groot, M., Overmars, M.: New results on binary space partitions in the plane. Comput. Geom. Theory Appl. 8, 317–333 (1997) MATHGoogle Scholar
  9. 9.
    de Berg, M., David, H., Katz, M., Overmars, M., van der Stappen, A.F., Vleugels, J.: Guarding scenes against invasive hypercubes. Comput. Geom. Theory Appl. 26, 99–117 (2003) MATHGoogle Scholar
  10. 10.
    de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications, 3rd edn. Springer, Berlin (2008) MATHGoogle Scholar
  11. 11.
    Chazelle, B., Edelsbrunner, H., Grigni, M., Guibas, L.J., Hershberger, J., Sharir, M., Snoeyink, J.: Ray shooting in polygons using geodesic triangulations. Algorithmica 12, 54–68 (1994) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Chin, N., Feiner, S.: Fast object-precision shadow generation for areal light sources using BSP trees. Comput. Graph. 25, 21–30 (1992) Google Scholar
  13. 13.
    Dumitrescu, A., Mitchell, J.S.B., Sharir, M.: Binary space partitions for axis-parallel segments, rectangles, and hyperrectangles. Discrete Comput. Geom. 31(2), 207–227 (2004) MATHMathSciNetGoogle Scholar
  14. 14.
    Fuchs, H., Kedem, Z.M., Naylor, B.: On visible surface generation by a priori tree structures. Comput. Graph. 14(3), 124–133 (1980). Proc. SIGGRAPH CrossRefGoogle Scholar
  15. 15.
    Hershberger, J., Suri, S.: A pedestrian approach to ray shooting: Shoot a ray, take a walk. J. Algorithms 18(3), 403–431 (1995) CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Hershberger, J., Suri, S., Tóth, Cs.D.: Binary space partitions of orthogonal subdivisions. SIAM J. Comput. 34(6), 1380–1397 (2005) CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Ishaque, M., Speckmann, B., Tóth, Cs.D.: Shooting permanent rays among disjoint polygons in the plane. In: Proc. 25th ACM Sympos. Comput. Geom., pp. 51–60. ACM Press, New York (2009) CrossRefGoogle Scholar
  18. 18.
    Mata, C.S., Mitchell, J.S.B.: Approximation algorithms for geometric tour and network design problems. In: Proc. 11th ACM Sympos. Comput. Geom., pp. 360–369. ACM Press, New York (1995) Google Scholar
  19. 19.
    Naylor, B.: Constructing good partitioning trees. In: Proc. Graphics Interface, pp. 181–191. Canadian Human-Computer Communications Society, Toronto (1993) Google Scholar
  20. 20.
    Paterson, M.S., Yao, F.F.: Efficient binary space partitions for hidden-surface removal and solid modeling. Discrete Comput. Geom. 5, 485–503 (1990) CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Paterson, M.S., Yao, F.F.: Optimal binary space partitions for orthogonal objects. J. Algorithms 13, 99–113 (1992) CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Schumacker, R.A., Brand, R., Gilliland, M., Sharp, W.: Study for applying computer-generated images to visual simulation. Tech. Rep. AFHRL-TR-69-14, San Antonio, TX (1969) Google Scholar
  23. 23.
    Thibault, W.C., Naylor, B.F.: Set operations on polyhedra using binary space partitioning trees. Comput. Graph. 21(4), 153–162 (1987). Proc. SIGGRAPH ’87 CrossRefMathSciNetGoogle Scholar
  24. 24.
    Tóth, Cs.D.: A note on binary plane partitions. Discrete Comput. Geom. 30, 3–16 (2003) MATHMathSciNetGoogle Scholar
  25. 25.
    Tóth, Cs.D.: Binary space partition for line segments with a limited number of directions. SIAM J. Comput. 32(2), 307–325 (2003) CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Tóth, Cs.D.: Binary space partition for axis-aligned fat rectangles. SIAM J. Comput. 38(1), 429–447 (2008) CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of CalgaryCalgaryCanada

Personalised recommendations