Skip to main content
SpringerLink
Log in
Book cover

Workshop on Algorithms and Data Structures

WADS 2003: Algorithms and Data Structures pp 219–230Cite as

Multi-way Space Partitioning Trees

  • Conference paper

  • 940 Accesses

Part of the book series: Lecture Notes in Computer Science ((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.

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

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  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. 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)

    Article  MATH  MathSciNet  Google Scholar 

  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. 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. 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)

    Article  Google Scholar 

  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. 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. 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. de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry Algorithms and Applications. Springer, Heidelberg (1997)

    MATH  Google Scholar 

  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. Foley, J.D., van Dam, A., Feiner, S.K., Hughes, J.F.: Computer Graphics: Principles and Practice. Addison-Wesley, Reading (1990)

    Google Scholar 

  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)

    Article  Google Scholar 

  13. Naylor, B., Amanatides, J.A., Thibault, W.: Merging BSP trees yields polyhedral set operations. Comp. Graph (SIGGRAPH 1990) 24(4), 115–124 (1990)

    Article  Google Scholar 

  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. Paterson, M.S., Yao, F.F.: Efficient binary space partitions for hidden-surface removal and solid modeling. Discrete Comput. Geom. 5, 485–503 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  16. Paterson, M.S., Yao, F.F.: Optimal binary space partitions for orthogonal objects. J. Algorithms 13, 99–113 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  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. Teller, S.J., Séquin, C.H.: Visibility preprocessing for interactive walkthroughs. Comput. Graph. 25(4), 61–69 (1991); Proc. SIGGRAPH 1991

    Article  Google Scholar 

  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 1987

    Article  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  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 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of Miami,  

    Christian A. Duncan

Authors
  1. Christian A. Duncan
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Carleton University, Ottawa, Canada

    Frank Dehne

  2. School of Computer Science, Carleton University, 1125 Colonel By Drive, K1S 5B6, Ottawa, Canada

    Jörg-Rüdiger Sack

  3. School of Computer Science, Carleton University, K1S 5B6, Ottawa, ON, Canada

    Michiel Smid

Rights and permissions

Reprints and permissions

Copyright information

© 2003 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Duncan, C.A. (2003). Multi-way Space Partitioning Trees. In: Dehne, F., Sack, JR., Smid, M. (eds) Algorithms and Data Structures. WADS 2003. Lecture Notes in Computer Science, vol 2748. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45078-8_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-45078-8_20

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40545-0

  • Online ISBN: 978-3-540-45078-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation