eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-06-01
55:1
55:16
10.4230/LIPIcs.SoCG.2022.55
article
Minimum Height Drawings of Ordered Trees in Polynomial Time: Homotopy Height of Tree Duals
Ophelders, Tim
1
2
https://orcid.org/0000-0002-9570-024X
Parsa, Salman
3
https://orcid.org/0000-0002-8179-9322
Department of Information and Computing Science, Utrecht University, The Netherlands
Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Scientific Computing and Imaging Institute, University of Utah, Salt Lake City, UT, USA
We consider drawings of graphs in the plane in which vertices are assigned distinct points in the plane and edges are drawn as simple curves connecting the vertices and such that the edges intersect only at their common endpoints. There is an intuitive quality measure for drawings of a graph that measures the height of a drawing ϕ : G↪ℝ² as follows. For a vertical line 𝓁 in ℝ², let the height of 𝓁 be the cardinality of the set 𝓁 ∩ ϕ(G). The height of a drawing of G is the maximum height over all vertical lines. In this paper, instead of abstract graphs, we fix a drawing and consider plane graphs. In other words, we are looking for a homeomorphism of the plane that minimizes the height of the resulting drawing. This problem is equivalent to the homotopy height problem in the plane, and the homotopic Fréchet distance problem. These problems were recently shown to lie in NP, but no polynomial-time algorithm or NP-hardness proof has been found since their formulation in 2009. We present the first polynomial-time algorithm for drawing trees with optimal height. This corresponds to a polynomial-time algorithm for the homotopy height where the triangulation has only one vertex (that is, a set of loops incident to a single vertex), so that its dual is a tree.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol224-socg2022/LIPIcs.SoCG.2022.55/LIPIcs.SoCG.2022.55.pdf
Graph drawing
homotopy height