Advertisement

Upward drawing on the plane grid using less ink

  • Guy-Vincent Jourdan
  • Ivan Rival
  • Nejib Zaguia
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 894)

Abstract

Any upward drawing D(P) on a two-dimensional integer grid I, of an ordered set P, has completion ¯P with an upward drawing D(¯P) on a two-dimensional integer grid Ī such that the total edge length of D(¯P) does not exceed the total edge length of D(P). Moreover, by (possibly) translating vertices, there is an upward drawing D(P) on I such that Ī=I.

Thus, any integer grid embedding of a two-dimensional ordered set can be extended to a planar upward drawing of its completion, on the same integer grid, without increasing the total edge length.

Keywords

Edge Length Minimum Span Tree Minkowski Plane Normal Completion Minimum Steiner Tree 
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.

References

  1. 1.
    A. Aeschlimann and J. Schmid (1992) Drawing orders using less ink, ORDER 9, 5–13.Google Scholar
  2. 2.
    R. Franzosa, L. Perry, A. Saalfeld, and A. Wohlgemuth (1994) An output-sensitive polynomial-time algorithm for constructing the normal completion of a partially ordered set, preprint.Google Scholar
  3. 3.
    B. Ganter and K. Reuter (1991) Finding all closed sets: a general approach, ORDER 8, 283–290.Google Scholar
  4. 4.
    B. Gao, D.-Z. Du, and R. L. Graham, The tight lower bound for the Steiner Ratio in Minkowski planes.Google Scholar
  5. 5.
    M. R. Garey and D. S. Johnson (1979) Computers and Intractability: A guide to the Theory of NP-Completeness, Freeman, x+338.Google Scholar
  6. 6.
    G.-V. Jourdan, J.-X. Rampon, and C. Jard (1994) Computing on-line the lattice of maximal antichains of posets, ORDER (to appear).Google Scholar
  7. 7.
    J. G. Lee, W.-P. Liu, R. Nowakowski, and I. Rival (1988) Dimension invariance of subdivision, Technical Report TR-88-30, University of Ottawa.Google Scholar
  8. 8.
    D. Kelly (1977) The 3-irreducible partially ordered sets, Canad. J. Math. 29, 367–383.Google Scholar
  9. 9.
    H. M. MacNeille (1937) Partially ordered sets, Trans. Amer. Math. Soc. 42, 416–460.Google Scholar
  10. 10.
    S. K. Rao, P. Sadayappan, F. K. Hwang, and P. W. Shor (1990) The rectilinear Steiner Arborescence Problem.Google Scholar
  11. 11.
    E. R. Tufte (1983) The Visual Display of Quantitative Information, Graphics Press, pp. 197.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1995

Authors and Affiliations

  • Guy-Vincent Jourdan
    • 1
  • Ivan Rival
    • 2
  • Nejib Zaguia
    • 2
  1. 1.IRISARennes CédexFrance
  2. 2.Department of Computer ScienceUniversity of OttawaOttawaCanada

Personalised recommendations