Advertisement

Multi-way Space Partitioning Trees

  • Christian A. Duncan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2748)

Abstract

In this paper, we introduce a new data structure, the multi-way space partitioning (MSP) tree similar in nature to the standard binary space partitioning (BSP) tree. Unlike the super-linear space requirement for BSP trees, we show that for any set of disjoint line segments in the plane there exists a linear-size MSP tree completely partitioning the set. Since our structure is a deviation from the standard BSP tree construction, we also describe an application of our algorithm. We prove that the well-known Painter’s algorithm can be adapted quite easily to use our structure to run in O(n) time. More importantly, the constant factor behind our tree size is extremely small, having size less than 4n.

Keywords

Line Segment Computational Geometry Rooted Segment Convex Region Binary Space 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Agarwal, P., Murali, T., Vitter, J.: Practical techniques for constructing binary space partitions for orthogonal rectangles. In: Proc. of the 13th Symposium on Computational Geometry, June 4-6, pp. 382–384. ACM Press, New York (1997)Google Scholar
  2. 2.
    Agarwal, P.K., Grove, E.F., Murali, T.M., Vitter, J.S.: Binary space partitions for fat rectangles. SIAM Journal on Computing 29(5), 1422–1448 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Airey, J.M.: Increasing Update Rates in the Building Walkthrough System with Automatic Model-Space Subdivision and Potentially Visible Set Calculations. PhD thesis, Dept. of CS, U. of North Carolina (July 1990) TR90-027 Google Scholar
  4. 4.
    Chin, N., Feiner, S.: Near real-time shadow generation using BSP trees. Computer Graphics (SIGGRAPH 1990 Proceedings) 24(4), 99–106 (1990)Google Scholar
  5. 5.
    Chin, N., Feiner, S.: Fast object-precision shadow generation for areal light sources using BSP trees. Computer Graphics (1992 Symposium on Interactive 3D Graphics) 25(4), 21–30 (1992)CrossRefGoogle Scholar
  6. 6.
    de Berg, de Groot, Overmars: New results on binary space partitions in the plane. CGTA: Computational Geometry: Theory and Applications 8 (1997)Google Scholar
  7. 7.
    de Berg, M.: Linear size binary space partitions for fat objects. In: Spirakis, P.G. (ed.) ESA 1995. LNCS, vol. 979, pp. 252–263. Springer, Heidelberg (1995)Google Scholar
  8. 8.
    de Berg, M., de Groot, M.: Binary space partitions for sets of cubes. In: Abstracts 10th European Workshop Comput. Geom., pp. 84–88 (1994)Google Scholar
  9. 9.
    de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry Algorithms and Applications. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  10. 10.
    Dumitrescu, A., Mitchell, J.S.G., Sharir, M.: Binary space partitions for axis-parallel segments, rectangles, and hyperrectangles. In: Proceedings of the 17th annual symposium on Computational geometry, pp. 141–150. ACM Press, New York (2001)Google Scholar
  11. 11.
    Foley, J.D., van Dam, A., Feiner, S.K., Hughes, J.F.: Computer Graphics: Principles and Practice. Addison-Wesley, Reading (1990)Google Scholar
  12. 12.
    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 1980 (1980) CrossRefGoogle Scholar
  13. 13.
    Naylor, B., Amanatides, J.A., Thibault, W.: Merging BSP trees yields polyhedral set operations. Comp. Graph (SIGGRAPH 1990) 24(4), 115–124 (1990)CrossRefGoogle Scholar
  14. 14.
    Naylor, B., Thibault, W.: Application of BSP trees to ray-tracing and CGS evaluation. Technical Report GIT-ICS 86/03, Georgia Institute of Tech., School of Information and Computer Science (February 1986) Google Scholar
  15. 15.
    Paterson, M.S., Yao, F.F.: Efficient binary space partitions for hidden-surface removal and solid modeling. Discrete Comput. Geom. 5, 485–503 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Paterson, M.S., Yao, F.F.: Optimal binary space partitions for orthogonal objects. J. Algorithms 13, 99–113 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Teller, S.J.: Visibility Computations in Densely Occluded Polyhedral Environments. PhD thesis, Dept. of Computer Science, University of California, Berkeley (1992) Google Scholar
  18. 18.
    Teller, S.J., Séquin, C.H.: Visibility preprocessing for interactive walkthroughs. Comput. Graph. 25(4), 61–69 (1991); Proc. SIGGRAPH 1991CrossRefGoogle Scholar
  19. 19.
    Thibault, W.C., Naylor, B.F.: Set operations on polyhedra using binary space partitioning trees. Comput. Graph. 21(4), 153–162 (1987); Proc. SIGGRAPH 1987CrossRefMathSciNetGoogle Scholar
  20. 20.
    Tóth, C.D.: A note on binary plane partitions. In: Proceedings of the seventeenth annual symposium on Computational geometry, pp. 151–156. ACM Press, New York (2001)CrossRefGoogle Scholar
  21. 21.
    Tóth, C.D.: Binary space partitions for line segments with a limited number of directions. In: Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms, pp. 465–471. ACM Press, New York (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Christian A. Duncan
    • 1
  1. 1.Department of Computer ScienceUniversity of Miami 

Personalised recommendations