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Acta Informatica

, Volume 42, Issue 6–7, pp 419–428 | Cite as

Embedding linear orders in grids

  • Andrzej Ehrenfeucht
  • Tero HarjuEmail author
  • Grzegorz Rozenberg
Original Article
  • 30 Downloads

Abstract

A grid (or a mesh) is a two-dimensional permutation: an m× n-grid of size mn is an m× n-matrix where the entries run through the elements {1,2, …, mn}. We prove that if δ1 and δ2 are any two linear orders on {1,2, …, N}, then they can be simultaneously embedded (in a well defined sense) into a unique grid having the smallest size.

Keywords

Linear Order Comparability Graph Order Preserve Permutation Graph Compatible Pair 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Baker, K.A., Fishburn, P.C., Roberts, F.S.: A new characterization of partial orders of dimension two. Ann. New York Acad. Sci. 175, 23–24 (1970)Google Scholar
  2. 2.
    Baker, K.A., Fishburn, P.C., Roberts, F.S.: Partial orders of dimension 2. Networks 2, 11–28 (1972)Google Scholar
  3. 3.
    Dushnik, B., Miller, E.W.: Partially ordered sets. Amer. J. Math. 63, 600–610 (1941)Google Scholar
  4. 4.
    Ehrenfeucht, A., Harju, T., Rozenberg, G.: The Theory of 2-Structures, World Scientific, Singapore (1999)Google Scholar
  5. 5.
    Ehrenfeucht, A., Harju, T., ten Pas, P., Rozenberg, G.: Permutations, parenthesis words, and Schröder numbers. Discrete Math. 190, 259–264 (1998)Google Scholar
  6. 6.
    Ehrenfeucht, A., Rozenberg, G.: T-structures, T-functions, and texts. Theoret. Comput. Sci. 116, 227–290 (1993)CrossRefGoogle Scholar
  7. 7.
    Ehrenfeucht, A., ten Pas, P., Rozenberg, G.: Combinatorial properties of texts. RAIRO Inform. Théor. Appl. 27, 433–464 (1993)Google Scholar
  8. 8.
    Pnueli, A., Lempel, A., Even, S.: Transitive orientation of graphs and identification of permutation graphs. Canad. J. Math. 23, 160–175 (1971)Google Scholar
  9. 9.
    Trotter, W.T.: Combinatorics and Partially Ordered Sets. Dimension Theory, The Johns Hopkins Univ. Press, Baltimore (1992)Google Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  • Andrzej Ehrenfeucht
    • 1
  • Tero Harju
    • 2
    Email author
  • Grzegorz Rozenberg
    • 1
    • 3
  1. 1.Department of Computer ScienceUniversity of Colorado at BoulderBoulderUSA
  2. 2.Department of MathematicsUniversity of TurkuTurkuFinland
  3. 3.Leiden Institute for Advanced Computer ScienceLeiden UniversityLeidenThe Netherlands

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