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Improved Bounds on the Union Complexity of Fat Objects

  • Conference paper
FSTTCS 2005: Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3821))

Abstract

We introduce a new class of fat, not necessarily convex or polygonal, objects in the plane, namely locally γ-fat objects. We prove that the union complexity of any set of n such objects is O(λ s + 2(n)log2 n). This improves the best known bound, and extends it to a more general class of objects.

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References

  1. Alt, H., Fleischer, R., Kaufmann, M., Mehlhorn, K., Naher, S., Schirra, S., Uhrig, C.: Approximate motion planning and the complexity of the boundary of the union of simple geometric figures. Algorithmica 8, 391–406 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  2. Aronov, B., Efrat, A., Koltun, V., Sharir, M.: On the union of κ-curved objects in three and four dimensions. In: Proc. 20th ACM Symp. Comput. Geom., pp. 383–390 (2004)

    Google Scholar 

  3. de Berg, M.: Vertical ray shooting for fat objects. In: Proc. 21st ACM Symp. Comput. Geom., pp. 288–295 (2005)

    Google Scholar 

  4. de Berg, M., Katz, M., van der Stappen, F., Vleugels, J.: Realistic input models for geometric algorithms. Algorithmica 34, 81–97 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  5. de Berg, M., van Kreveld, M., Overmars, M., Schwarzkopf, O.: Computational Geometry: Algorithms and Applications, 2nd edn. Springer, Heidelberg (2000)

    MATH  Google Scholar 

  6. Duncan, C.A.: Balanced Aspect Ratio Trees. Ph.D. Thesis, John Hopkins University (1999)

    Google Scholar 

  7. Duncan, C.A., Goodrich, M.T., Kobourov, S.G.: Balanced aspect ratio trees: Combining the advantages of k-d trees and octrees. In: Proc. 10th Ann. ACM-SIAM Sympos. Discrete Algorithms, pp. 300–309 (1999)

    Google Scholar 

  8. Efrat, A.: The complexity of the union of (α, β)-covered objects. SIAM J. Comput. 34, 775–787 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Efrat, A., Rote, G., Sharir, M.: On the union of fat wedges and separating a collection of segments by a line. Comput. Geom. Theory Appl. 3, 277–288 (1993)

    MATH  MathSciNet  Google Scholar 

  10. Efrat, A., Sharir, M.: The complexity of the union of fat objects in the plane. In: Proc. 13th ACM Symp. Comput. Geom., pp. 104–112 (1997)

    Google Scholar 

  11. Efrat, A., Katz, M.: On the union of α-curved objects. In: Proc. 14th ACM Symp. Comput. Geom., pp. 206–213 (1998)

    Google Scholar 

  12. Katz, M.J., Overmars, M., Sharir, M.: Efficient output sensitive hidden surface removal for objects with small union size. Comput. Geom. Theory Appl. 2, 223–234 (1992)

    MATH  MathSciNet  Google Scholar 

  13. Katz, M.J.: 3-D vertical ray shooting and 2-D point enclosure, range searching, and arc shooting amidst convex fat objects. Comput. Geom. Theory Appl. 8, 299–316 (1998)

    Google Scholar 

  14. Kedem, K., Livne, R., Pach, J., Sharir, M.: On the union of Jordan regions and collision-free translational motion amidst polygonal obstacles in the plane. Comput. Geom. 1, 59–71 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  15. van Kreveld, M.: On fat partitioning, fat covering, and the union size of polygons. Comput. Geom. Theory Appl. 9, 197–210 (1998)

    MATH  Google Scholar 

  16. Matoušek, J., Pach, J., Sharir, M., Sifrony, S., Welzl, E.: Fat triangles determine linearly many holes. SIAM J. Comput. 23, 154–169 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  17. Pach, J., Safruti, I., Sharir, M.: The union of congruent cubes in three dimensions. Discr. Comput. Geom. 30, 133–160 (2003)

    MATH  MathSciNet  Google Scholar 

  18. Pach, J., Tardos, G.: On the boundary complexity of the union of fat triangles. SIAM J. Comput. 31, 1745–1760 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Sharir, M., Agarwal, P.K.: Davenport-Schinzel sequences and their geometric applications. Cambridge University Press, Cambridge (1995)

    MATH  Google Scholar 

  20. van der Stappen, A.F.: Motion planning amidst fat obstacles. Ph.D. thesis, Utrecht University, Utrecht, the Netherlands (1994)

    Google Scholar 

  21. van der Stappen, A.F., Halperin, D., Overmars, M.H.: The complexity of the free space for a robot moving amidst fat obstacles. Comput. Geom. Theory Appl. 3, 353–373 (1993)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Computing Science, TU Eindhoven, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    Mark de Berg

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  1. Mark de Berg
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Editor information

Editors and Affiliations

  1. National Institute of Advanced Studies, Indian Institute of Science Campus, 560012, Bangalore, India

    Sundar Sarukkai

  2. Department of Comp. Sc. & Engg., Indian Institute of Technology Delhi, 110016, New Delhi, India

    Sandeep Sen

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© 2005 Springer-Verlag Berlin Heidelberg

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de Berg, M. (2005). Improved Bounds on the Union Complexity of Fat Objects. In: Sarukkai, S., Sen, S. (eds) FSTTCS 2005: Foundations of Software Technology and Theoretical Computer Science. FSTTCS 2005. Lecture Notes in Computer Science, vol 3821. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11590156_9

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  • DOI: https://doi.org/10.1007/11590156_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-30495-1

  • Online ISBN: 978-3-540-32419-5

  • eBook Packages: Computer ScienceComputer Science (R0)

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