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Scandinavian Workshop on Algorithm Theory

SWAT 2004: Algorithm Theory - SWAT 2004 pp 114–126Cite as

Matching Polyhedral Terrains Using Overlays of Envelopes

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 3111))

Abstract

We show that the combinatorial complexity of the overlay of the lower envelopes of two collections of d-variate piecewise linear functions of overall combinatorial complexity n is Ω(n d α 2(n)) and O(n d + ε) for any ε> 0 when d ≥ 2, and O(n 2 α(n) log n) when d=2. This extends and improves the analysis of de Berg et al. [9]. We also describe an algorithm that constructs the overlay in the same time.

We apply these results to obtain efficient general solutions to the problem of matching two polyhedral terrains in ℝd + 1 under translation. For the perpendicular distance measure, which we adopt from functional analysis, we present a matching algorithm that runs in time O(n 2d + ε) for any ε> 0. For the directed and undirected Hausdorff distance measures, we present a matching algorithm that runs in time O(n \(^{d^2+d+\epsilon}\)) for any ε> 0.

A limited preliminary version of some of the results described in this paper has appeared in the second author’s Ph.D. thesis [22].

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References

  1. Agarwal, P.K., Har-Peled, S., Sharir, M., Wang, Y.: Hausdorff distance under translation for points, disks, and balls. In: Proc. Symp. Comp. Geom (2003)

    Google Scholar 

  2. Agarwal, P.K., Schwarzkopf, O., Sharir, M.: The overlay of lower envelopes and its applications. Discrete and Computational Geometry 15, 1–13 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  3. Agarwal, P.K., Sharir, M.: Arrangements and their applications. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 49–119. Elsevier Science Publishers B.V. North-Holland, Amsterdam (2000)

    Chapter  Google Scholar 

  4. Alt, H., Guibas, L.: Discrete geometric shapes: Matching, interpolation, and approximation – a survey. In: Sack, J.-R., Urrutia, J. (eds.) Handbook of Computational Geometry, Elsevier Science Publishers B.V. North-Holland (2000)

    Google Scholar 

  5. Boissonnat, J.-D., Dobrindt, K.: On-line construction of the upper envelope of triangles and surface patches in three dimensions. Comput. Geom. Theory Appl. 5, 303–320 (1996)

    MATH  MathSciNet  Google Scholar 

  6. Boissonnat, J.-D., Yvinec, M.: Algorithmic Geometry. Cambridge University Press, Cambridge (1998)

    MATH  Google Scholar 

  7. Chazelle, B.: Cutting hyperplanes for divide-and-conquer. Discrete Comput. Geom. 9(2), 145–158 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  8. Clarkson, K.L.: A randomized algorithm for closest-point queries. SIAM Journal on Computing 17, 830–847 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  9. de Berg, M., Guibas, L.J., Halperin, D.: Vertical decompositions for triangles in 3-space. Discrete Comput. Geom. 15, 35–61 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. Edelsbrunner, H.: The upper envelope of piecewise linear functions: Tight complexity bounds in higher dimensions. Discrete and Comp. Geom. 4, 337–343 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  11. Edelsbrunner, H., Guibas, L., Sharir, M.: The upper envelope of piecewise linear functions: algorithms and applications. Discr. Comp. Geom. 4, 311–336 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  12. Hershberger, J.: Finding the upper envelope of n line segments in O(n log n) time. Inform. Process. Lett. 33, 169–174 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  13. Koltun, V., Sharir, M.: The partition technique for overlays of envelopes. SIAM Journal on Computing (to appear)

    Google Scholar 

  14. Koltun, V., Wenk, C.: Matching polyhedral terrains using overlays of envelopes, http://www.cs.berkeley.edu/~vladlen/linear-overlays-full.zip

  15. Lee, C.: Subdivisions and triangulations of polytopes. In: Goodman, J.E., O’Rourke, J. (eds.) Discr. Comp. Geom, pp. 271–290. CRC Press, Boca Raton (1997)

    Google Scholar 

  16. Matoušek, J.: Range searching with efficient hierarchical cuttings. Discrete Comput. Geom. 10(2), 157–182 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  17. Megiddo, N.: Applying parallel computation algorithms in the design of serial algorithms. J. ACM 30(4), 852–865 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  18. Pach, J., Sharir, M.: The upper envelope of piecewise linear functions and the boundary of a region enclosed by convex plates: combinatorial analysis. Discrete Comput. Geom. 4, 291–309 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  19. Sharir, M.: Almost tight upper bounds for lower envelopes in higher dimensions. Discrete and Computational Geometry 12, 327–345 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  20. Sharir, M., Agarwal, P.K.: Davenport-Schinzel Sequences and Their Geometric Applications. Cambridge University Press, New York (1995)

    MATH  Google Scholar 

  21. Tagansky, B.: A new technique for analyzing substructures in arrangements of piecewise linear surfaces. Discrete and Computational Geometry 16, 455–479 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  22. Wenk, C.: Shape Matching in Higher Dimensions. Ph.D. thesis, Free University Berlin, Berlin, Germany (2002)

    Google Scholar 

  23. Wiernik, A., Sharir, M.: Planar realizations of nonlinear Davenport-Schinzel sequences by segments. Discrete Comput. Geom. 3, 15–47 (1988)

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Computer Science Division, University of California, Berkeley, CA, 94720-1776, USA

    Vladlen Koltun

  2. Dept. of Computer Science, University of Texas at San Antonio, 6900 N. Loop 1604 West, San Antonio, TX, 78249-0667, USA

    Carola Wenk

Authors
  1. Vladlen Koltun
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  2. Carola Wenk
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Editor information

Editors and Affiliations

  1. Institut für Informatik, Universität Augsburg, 86135, Augsburg, Germany

    Torben Hagerup

  2. Department of Computing, University of Copenhagen, Denmark

    Jyrki Katajainen

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Koltun, V., Wenk, C. (2004). Matching Polyhedral Terrains Using Overlays of Envelopes. In: Hagerup, T., Katajainen, J. (eds) Algorithm Theory - SWAT 2004. SWAT 2004. Lecture Notes in Computer Science, vol 3111. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-27810-8_11

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  • DOI: https://doi.org/10.1007/978-3-540-27810-8_11

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-22339-9

  • Online ISBN: 978-3-540-27810-8

  • eBook Packages: Springer Book Archive

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