Minimum Height Drawings of Ordered Trees in Polynomial Time: Homotopy Height of Tree Duals

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 $φ: G \rightarrow \mathbb{R}^2$ as follows. For a vertical line $\ell$ in $\mathbb{R}^2$, let the height of $\ell$ be the cardinality of the set $\ell \cap φ(G)$. 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.