Skip to main content
Springer Nature Link
Log in

On the Parameterized Complexity of d-Dimensional Point Set Pattern Matching

  • Conference paper
Parameterized and Exact Computation (IWPEC 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4169))

Included in the following conference series:

  • 832 Accesses

Abstract

Deciding whether two n-point sets A, B ∈ℝd are congruent is a fundamental problem in geometric pattern matching. When the dimension d is unbounded, the problem is equivalent to graph isomorphism and is conjectured to be in FPT.

When |A|=m<|B|=n, the problem becomes that of deciding whether A is congruent to a subset of B and is known to be NP-complete. We show that point subset congruence, with d as a parameter, is W[1]-hard, and that it cannot be solved in O(mn o(d))-time, unless SNP⊂DTIME(2o(n)). This shows that, unless FPT=W[1], the problem of finding an isometry of A that minimizes its directed Hausdorff distance, or its Earth Mover’s Distance, to B, is not in FPT.

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

Access this chapter

Log in via an institution

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Akutsu, T.: On determining the congruence of point sets in d-dimensions. Comput. Geom. Theory Appl. 9(4), 247–256 (1998)

    MATH  MathSciNet  Google Scholar 

  2. Alt, H., Guibas, L.J.: Discrete geometric shapes: Matching, interpolation, and approximation. In: Sack, J.R., Urrutia, J. (eds.) Handbook of Computational Geometry, pp. 121–153. Elsevier Science Publishers B.V., North-Holland (1999)

    Google Scholar 

  3. Alt, H., Mehlhorn, K., Wagener, H., Welzl, E.: Congruence, similarity, and symmetries of geometric objects. Discrete Comput. Geom. 3, 237–256 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  4. Braß, P.: Exact point pattern matching and the number of congruent triangles in a three-dimensional pointset. In: Paterson, M. (ed.) ESA 2000. LNCS, vol. 1879, pp. 112–119. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  5. Braß, P., Knauer, C.: Testing the congruence of d-dimensional point sets. Int. J. of Comp. Geom. Appl. 12(1), 115–124 (2002)

    Article  MATH  Google Scholar 

  6. Braß, P., Pach, J.: Problems and results on geometric patterns. In: Avis, D., Hertz, A., Marcotte, O. (eds.) Graph Theory and Combinatorial Optimization, Gerad 25th Anniversary, pp. 17–36. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  7. Cabello, S., Giannopoulos, P., Knauer, C., Rote, G.: Matching point sets with respect to the earth mover’s distance. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 520–531. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  8. Chen, J., Chor, B., Fellows, M., Huang, X., Juedes, D., Kanj, I.A., Xia, G.: Tight lower bounds for certain parameterized NP-hard problems. Information and Computation 201(2), 216–231 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chew, L.P., Dor, D., Efrat, A., Kedem, K.: Geometric pattern matching in d-dimensional space. Discrete Comput. Geom. 21 (1999)

    Google Scholar 

  10. Chew, L.P., Goodrich, M., Huttenlocher, D.P., Kedem, K., Kleinberg, J.M., Kravets, D.: Geometric pattern matching under Euclidean motion. Comput. Geom. Theory Appl. 7, 113–124 (1997)

    MATH  MathSciNet  Google Scholar 

  11. Cohen, S.D., Guibas, L.J.: The Earth Mover’s Distance under transformation sets. In: Proceedings of the 7th IEEE International Conference on Computer Vision, pp. 173–187 (September 1999)

    Google Scholar 

  12. de Rezende, P.J., Lee, D.T.: Point set matching in d-dimensions. Algorithmica 13, 387–404 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Efrat, A., Itai, A., Katz, M.J.: Geometry helps in bottleneck matching and related problems. Algorithmica 31, 1–28 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Flum, J., Grohe, M.: Parameterized Complexity Theory. Texts in Theoretical Computer Science. An EATCS Series, vol. XIV. Springer, Heidelberg (2006)

    Google Scholar 

  15. Hagedoorn, M., Veltkamp, R.C.: State-of-the-art in shape matching. Technical Report UU-CS-1999-027, Institute of Information and Computing Sciences, Utrecht University, The Netherlands (1999)

    Google Scholar 

  16. Huttenlocher, D.P., Kedem, K., Sharir, M.: The upper envelope of Voronoi surfaces and its applications. Discrete Comput. Geom. 9, 267–291 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  17. Marx, D.: Efficient approximation schemes for geometric problems? In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 448–459. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  18. Megiddo, N.: Linear programming in linear time when the dimension is fixed. J. ACM 31, 114–127 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  19. Megiddo, N.: On the complexity of some geometric problems in unbounded dimension. J. Symb. Comput. 10(3-4), 327–334 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  20. Papadimitriou, C., Safra, S.: The complexity of low-distortion embeddings between point sets. In: Proc. of the 16th Annu. ACM-SIAM Sympos. Discrete Algorithms (SODA), pp. 112–118. SIAM, Philadelphia (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, IMFM, Jadranska 19, SI-1000, Ljubljana, Slovenia

    Sergio Cabello

  2. Institut für Informatik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099, Berlin, Germany

    Panos Giannopoulos

  3. Institut für Informatik, Freie Universität Berlin, Takustraße 9, D-14195, Berlin, Germany

    Christian Knauer

Authors
  1. Sergio Cabello
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Panos Giannopoulos
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Christian Knauer
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Information and Computing Sciences, Utrecht University, The Netherlands

    Hans L. Bodlaender

  2. University of Tennessee, 37996-3450, Knoxville, Tennessee, U.S.A.

    Michael A. Langston

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Cabello, S., Giannopoulos, P., Knauer, C. (2006). On the Parameterized Complexity of d-Dimensional Point Set Pattern Matching. In: Bodlaender, H.L., Langston, M.A. (eds) Parameterized and Exact Computation. IWPEC 2006. Lecture Notes in Computer Science, vol 4169. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11847250_16

Download citation

  • DOI: https://doi.org/10.1007/11847250_16

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-39098-5

  • Online ISBN: 978-3-540-39101-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics