Upward drawing on the plane grid using less ink
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.
KeywordsEdge Length Minimum Span Tree Minkowski Plane Normal Completion Minimum Steiner Tree
- 1.A. Aeschlimann and J. Schmid (1992) Drawing orders using less ink, ORDER 9, 5–13.Google Scholar
- 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.B. Ganter and K. Reuter (1991) Finding all closed sets: a general approach, ORDER 8, 283–290.Google Scholar
- 4.B. Gao, D.-Z. Du, and R. L. Graham, The tight lower bound for the Steiner Ratio in Minkowski planes.Google Scholar
- 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.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.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.D. Kelly (1977) The 3-irreducible partially ordered sets, Canad. J. Math. 29, 367–383.Google Scholar
- 9.H. M. MacNeille (1937) Partially ordered sets, Trans. Amer. Math. Soc. 42, 416–460.Google Scholar
- 10.S. K. Rao, P. Sadayappan, F. K. Hwang, and P. W. Shor (1990) The rectilinear Steiner Arborescence Problem.Google Scholar
- 11.E. R. Tufte (1983) The Visual Display of Quantitative Information, Graphics Press, pp. 197.Google Scholar