Skip to main content
SpringerLink
Log in
Book cover

Latin American Symposium on Theoretical Informatics

LATIN 2004: LATIN 2004: Theoretical Informatics pp 80–89Cite as

Sampling Grid Colorings with Fewer Colors

  • Conference paper

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2976))

Abstract

We provide an optimally mixing Markov chain for 6-colorings of the square grid. Furthermore, this implies that the uniform distribution on the set of such colorings has strong spatial mixing. Four and five are now the only remaining values of k for which it is not known whether there exists a rapidly mixing Markov chain for k-colorings of the square grid.

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   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

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Aldous, D.: Random walks on finite groups and rapidly mixing Markov chains. In: Dold, A., Eckmann, B. (eds.) Séminaire de Probabilités XVII 1981/82. Springer Lecture Notes in Mathematics, vol. 986, pp. 243–297 (1986)

    Google Scholar 

  2. Bubley, R., Dyer, M.: Path coupling: a technique for proving rapid mixing in Markov chains. In: Proc. 28th Ann. Symp. on Found. of Comp. Sci., pp. 223–231 (1997)

    Google Scholar 

  3. Bubley, R., Dyer, M., Greenhill, C.: Beating the 2Δ bound for approximately counting colourings: A computer-assisted proof of rapid mixing. In: Proc. 9th Ann. ACM-SIAM Symposium on Discrete Algorithms, pp. 355–363 (1998)

    Google Scholar 

  4. Cesi, F.: Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields. Probability Theory and Related Fields 120, 569–584 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Diaconis, P., Saloff-Coste, L.: Comparison theorems for reversible Markov chains. Annals of Applied Probability 6, 696–730 (1996)

    MathSciNet  Google Scholar 

  6. Diaconis, P., Stroock, D.: Geometric bounds for eigenvalues of Markov chains. Annals of Applied Probability 1, 36–61 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dyer, M., Greenhill, C.: A more rapidly mixing Markov chain for graph colorings. Random Structures and Algorithms 13, 285–317 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dyer, M., Greenhill, C.: Random walks on combinatorial objects. In: Lamb, J., Preece, D. (eds.) Surveys in Combinatorics, pp. 101–136. Cambridge University Press, Cambridge (1999)

    Google Scholar 

  9. Dyer, M., Sinclair, A., Vigoda, E., Weitz, D.: Mixing in time and space for lattice spin systems: A combinatorial view. In: Rolim, J.D.P., Vadhan, S.P. (eds.) RANDOM 2002. LNCS, vol. 2483, pp. 149–163. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  10. Ferreira, S., Sokal, A.: Antiferromagnetic Potts models on the square lattice: a high-precision Monte Carlo study. J. Statistical Physics 96, 461–530 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Goldberg, L., Martin, R., Paterson, M.: Random sampling of 3-colourings in ℤ2. Random Structures and Algorithms (to appear)

    Google Scholar 

  12. Jerrum, M.: A very simple algorithm for estimating the number of k-colorings of a low-degree graph. Random Structures and Algorithms 7, 157–165 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Luby, M., Randall, D., Sinclair, A.: Markov chain algorithms for planar lattice structures. SIAM Computing 31, 167–192 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Martinelli, F.: Lectures on Glauber dynamics for discrete spin models. In: Lectures on Probability Theory and Statistics, Saint-Flour 1997. Springer Lecture Notes in Mathematics, vol. 1717, pp. 93–191 (1999)

    Google Scholar 

  15. Moore, C., Newman, M.: Height representation, critical exponents, and ergodicity in the four-state triangular Potts antiferromagnet. J. Stat. Phys. 99, 661–690 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Moore, C., Nordahl, M., Minar, N., Shalizi, C.: Vortex dynamics and entropic forces in antiferromagnets and antiferromagnetic Potts models. Physical Review E 60, 5344–5351 (1999)

    Article  Google Scholar 

  17. Randall, D., Tetali, P.: Analyzing Glauber dynamics by comparison of Markov chains. J. Mathematical Physics 41, 1598–1615 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  18. Salas, J., Sokal, A.: Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem. J. Statistical Physics 86, 551–579 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  19. Sinclair, A.: Algorithms for random generation and counting: a Markov chain approach, pp. 47–48. Birkhauser, Boston (1993)

    MATH  Google Scholar 

  20. Sinclair, A., Jerrum, M.: Approximate counting, uniform generation, and rapidly mixing Markov chains. Information and Computation 82, 93–133 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  21. Sokal, A.: A personal list of unsolved problems concerning lattice gases and antiferromagnetic Potts models. Talk presented at the conference on Inhomogeneous Random Systems, Université de Cergy-Pontoise (January 2000); Markov Processes and Related Fields 7, 21–38 (2001)

    Google Scholar 

  22. Vigoda, E.: Improved bounds for sampling colorings. J. Mathematical Physics 41, 1555–1569 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  23. Wang, J., Swendsen, R., Kotecký, R.: Physical Review Letters 63, 109 (1989)

    Article  Google Scholar 

  24. Wang, J., Swendsen, R., Kotecký, R.: Physical Review B 42, 2465 (1990)

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Microsoft Research, ‘

    Dimitris Achlioptas

  2. Dept of Computer Science, University of Toronto, and Microsoft Research, ‘

    Mike Molloy

  3. Computer Science Department, University of New Mexico, ‘

    Cristopher Moore

  4. Dept of Computer Science, University of Toronto, ‘

    Frank Van Bussel

Authors
  1. Dimitris Achlioptas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mike Molloy
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Cristopher Moore
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Frank Van Bussel
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, Rutgers University, NJ 08854, Piscataway

    Martín Farach-Colton

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Achlioptas, D., Molloy, M., Moore, C., Van Bussel, F. (2004). Sampling Grid Colorings with Fewer Colors. In: Farach-Colton, M. (eds) LATIN 2004: Theoretical Informatics. LATIN 2004. Lecture Notes in Computer Science, vol 2976. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24698-5_12

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-24698-5_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-21258-4

  • Online ISBN: 978-3-540-24698-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation