On Upward Drawings of Trees on a Given Grid

Computing a minimum-area planar straight-line drawing of a graph is known to be NP-hard for planar graphs, even when restricted to outerplanar graphs. However, the complexity question is open for trees. Only a few hardness results are known for straight-line drawings of trees under various restrictions such as edge length or slope constraints. On the other hand, there exist polynomial-time algorithms for computing minimum-width (resp., minimum-height) upward drawings of trees, where the height (resp., width) is unbounded. In this paper we take a major step in understanding the complexity of the area minimization problem for strictly-upward drawings of trees, which is one of the most common styles for drawing rooted trees. We prove that given a rooted tree $T$ and a $W\times H$ grid, it is NP-hard to decide whether $T$ admits a strictly-upward (unordered) drawing in the given grid.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Drawing Binary Tanglegrams: An Experimental Evaluation

A binary tanglegram is a pair of binary trees whose leaf sets are in one-to-one correspondence; matching leaves are connected by inter-tree edges. For applications, for example in phylogenetics or software engineering, it is required that the individual trees are drawn crossing-free. A natural optimization problem, denoted tanglegram layout problem, is thus to minimize the number of crossings between inter-tree edges. The tanglegram layout problem is NP-hard and is currently considered both in application domains and theory. In this paper we present an experimental comparison of a recursive algorithm of Buchin et al., our variant of their algorithm, the algorithm hierarchy sort of Holten and van Wijk, and an integer quadratic program that yields optimal solutions.Comment: see http://www.siam.org/proceedings/alenex/2009/alx09_011_nollenburgm.pd

Improved Bounds for Drawing Trees on Fixed Points with L-shaped Edges

Let $T$ be an $n$-node tree of maximum degree 4, and let $P$ be a set of $n$ points in the plane with no two points on the same horizontal or vertical line. It is an open question whether $T$ always has a planar drawing on $P$ such that each edge is drawn as an orthogonal path with one bend (an "L-shaped" edge). By giving new methods for drawing trees, we improve the bounds on the size of the point set $P$ for which such drawings are possible to: $O(n^{1.55})$ for maximum degree 4 trees; $O(n^{1.22})$ for maximum degree 3 (binary) trees; and $O(n^{1.142})$ for perfect binary trees. Drawing ordered trees with L-shaped edges is harder---we give an example that cannot be done and a bound of $O(n \log n)$ points for L-shaped drawings of ordered caterpillars, which contrasts with the known linear bound for unordered caterpillars.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017