Skip to main content
SpringerLink
Log in

Theory and Application of Width Bounded Geometric Separator

  • Conference paper
STACS 2006 (STACS 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3884))

Included in the following conference series:

Abstract

We introduce the notion of the width bounded geometric separator and develop the techniques for the existence of the width bounded separator in any d-dimensional Euclidean space. The separator is applied in obtaining \(2^{O(\sqrt{n})}\) time exact algorithms for a class of NP-complete geometric problems, whose previous algorithms take \(n^{O(\sqrt{n})}\) time[2][5][1]. One of those problems is the well known disk covering problem, which seeks to determine the minimal number of fixed size disks to cover n points on a plane[10]. They also include some NP-hard problems on disk graphs such as the maximum independent set problem, the vertex cover problem, and the minimum dominating set problem.

This research is supported by Louisiana Board of Regents fund under contract number LEQSF(2004-07)-RD-A-35.

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

Access this chapter

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
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

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Agarwal, P., Overmars, M., Sharir, M.: Computing maximally separated sets in the plane and independent sets in the intersection graph of unit graph. In: Proceedings of 15th ACM-SIAM symposium on discrete mathematics algorithms, pp. 509–518. ACM-SIAM (2004)

    Google Scholar 

  2. Agarwal, P., Procopiuc, C.M.: Exact and approximation algorithms for clustering. Algorithmica 33, 201–226 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  3. Alber, J., Fernau, H., Niedermeier, R.: Graph separators: a parameterized view. In: Proceedings of 7th Internal computing and combinatorics conference, pp. 318–327 (2001)

    Google Scholar 

  4. Alber, J., Fernau, H., Niedermeier, R.: Parameterized complexity: exponential speed-up for planar graph problems. In: Proceedings of 28st international colloquium on automata, languages and programming, pp. 261–272 (2001)

    Google Scholar 

  5. Alber, J., Fiala, J.: Geometric separation and exact solution for parameterized independent set problem on disk graphs. Journal of Algorithms 52(2), 134–151 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Alon, N., Seymour, P., Thomas, R.: Planar Separator. SIAM J. Discr. Math. 7(2), 184–193 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  7. Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete mathematics 86, 165–177 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  8. Djidjev, H.N.: On the problem of partitioning planar graphs. SIAM journal on discrete mathematics 3(2), 229–240 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  9. Djidjev, H.N., Venkatesan, S.M.: Reduced constants for simple cycle graph separation. Acta informatica 34, 231–234 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fowler, R.J., Paterson, M.S., Tanimoto, S.L.: Optimal packing and covering in the plane are NP-complete. Information processing letters 3(12), 133–137 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fu, B., Wang, W.: A \(2^{O(n^{1-1/d}\log n)}\)-time algorithm for d-dimensional protein folding in the HP-model. In: Proceedings of 31st international colloquium on automata, languages and programming, pp. 630–644 (2004)

    Google Scholar 

  12. Fu, B., Chen, Z.: Sublinear-time algorithms for width-bounded geometric separator and their application to protein side-chain packing problem (submitted)

    Google Scholar 

  13. Gazit, H.: An improved algorithm for separating a planar graph, manuscript, USC (1986)

    Google Scholar 

  14. Gilbert, J.R., Hutchinson, J.P., Tarjan, R.E.: A separation theorem for graphs of bounded genus. Journal of algorithm (5), 391–407 (1984)

    Article  MATH  Google Scholar 

  15. Graham, R., Grötschel, M., Lovász, L.: Handbook of combinatorics, vol. I. MIT Press, Cambridge (1996)

    MATH  Google Scholar 

  16. Hales, T.C.: A computer verification of the Kepler conjecture. In: Proceedings of the ICM, Beijing, vol. 3, pp. 795–804 (2002)

    Google Scholar 

  17. Lichtenstein, D.: Planar formula and their uses. SIAM journal on computing 11(2), 329–343 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lipton, R.J., Tarjan, R.: A separator theorem for planar graph. SIAM Journal on Applied Mathematics 36, 177–189 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lipton, R.J., Tarjan, R.: Applications of a planar separator theorem. SIAM journal on computing 9(3), 615–627 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  20. Meggido, N., Supowit, K.: On the complexity of some common geometric location problems. SIAM journal on computing 13, 1–29 (1984)

    MathSciNet  Google Scholar 

  21. Miller, G.L., Teng, S.-H., Vavasis, S.A.: An unified geometric approach to graph separators. In: 32nd annual symposium on foundation of computer science, pp. 538–547. IEEE, Los Alamitos (1991)

    Google Scholar 

  22. Miller, G.L., Thurston, W.: Separators in two and three dimensions. In: 22nd Annual ACM symposium on theory of computing, pp. 300–309. ACM, New York (1990)

    Google Scholar 

  23. Ravi, S.S., Hunt III, H.B.: Application of the planar separator theorem to computing problems. Information processing letter 25(5), 317–322 (1987)

    Article  MATH  Google Scholar 

  24. Smith, W.D., Wormald, N.C.: Application of geometric separator theorems. In: The 39th annual symposium on foundations of computer science, pp. 232–243 (1998)

    Google Scholar 

  25. Spielman, D.A., Teng, S.H.: Disk packings and planar separators. In: The 12th annual ACM symposium on computational geometry, pp. 349–358 (1996)

    Google Scholar 

  26. Pach, J., Agarwal, P.K.: Combinatorial geometry. Wiley-Interscience Publication, Chichester (1995)

    Book  MATH  Google Scholar 

  27. Wong, D.W., Kuo, Y.S.: A study of two geometric location problems. Information processing letters 28, 281–286 (1988)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dept. of Computer Science, University of New Orleans, LA, 70148, USA

    Bin Fu

  2. Research Institute for Children, 200 Henry Clay Avenue, LA, 70118, USA

    Bin Fu

Authors
  1. Bin Fu
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. LIF, CNRS & Univ. de Provence,, Marseille, France

    Bruno Durand

  2. RWTH Aachen University,, Lehrstuhl Informatik 7,, 52056,, Aachen,, Germany

    Wolfgang Thomas

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Fu, B. (2006). Theory and Application of Width Bounded Geometric Separator. In: Durand, B., Thomas, W. (eds) STACS 2006. STACS 2006. Lecture Notes in Computer Science, vol 3884. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11672142_22

Download citation

  • DOI: https://doi.org/10.1007/11672142_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-32301-3

  • Online ISBN: 978-3-540-32288-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation