Simultaneous Drawing of Layered Trees

TL;DR

This work tackles crossing minimization in layered drawings of forests of rooted trees with a fixed leaf order on layer $1$. For the case of two trees on any number of layers, it provides a polynomial-time dynamic program that preserves the prescribed planar embeddings and yields a minimum-crossing drawing, running in $O(n^3)$. For three layers with an arbitrary number of trees, it presents an XP-time algorithm in the number of trees by reducing the problem to a shortest-path computation on a $k$-dimensional grid graph, achieving $O(n^k)$ time. The paper further discusses extensions to planar graphs under certain conditions and outlines open questions, including the complexity for $k\ge3$ and $\ell\ge4$ and potential fixed-parameter tractability results. Overall, it delineates a clear boundary between tractable and parameterized cases for simultaneous layered tree drawings and provides practical methods for drawing multiple trees with minimal crossings.

Abstract

We study the crossing-minimization problem in a layered graph drawing of planar-embedded rooted trees whose leaves have a given total order on the first layer, which adheres to the embedding of each individual tree. The task is then to permute the vertices on the other layers (respecting the given tree embeddings) in order to minimize the number of crossings. While this problem is known to be NP-hard for multiple trees even on just two layers, we describe a dynamic program running in polynomial time for the restricted case of two trees. If there are more than two trees, we restrict the number of layers to three, which allows for a reduction to a shortest-path problem. This way, we achieve XP-time in the number of trees.

Figures (3)

• Figure 1: (a) Upward drawing of $k$ disjoint directed rooted trees $T_1,\dots T_k$ on $\ell$ layers. As indicated by the filled vertices, the total order $<_1$ of layer $1$ is given, while the total orders $<_2,\dots, <_\ell$ need to be determined. In the following figures, we drop the arrowheads and assume an upward direction. (b) Illustration of positions (gray boxes) with respect to $T_1$ and their respective ideal positions indicated by a directed gray arrow from each position $p$ to its ideal position $p^\star$.
• Figure 2: Example of a vertex $v$ of $V_2(T_2)$ having three children $c_1, c_2, c_3$, where the position $p_c$ of a child $c$ and the position $p$ of $v$ determine the value of $o[v,p]$. Here, we perceive $\mathop{\mathrm{\chi}}\nolimits$ and $o$ as functions dependent on $p$.
• Figure 3: Reducing the problem of finding a layered drawing of $k$ trees on three layers with the minimum number of crossings, where the leaves and the roots are fixed, to a shortest-path problem in a weighted $k$-dimensional grid graph.

Theorems & Definitions (22)

• theorem thmcountertheorem
• proof
• Claim 1.1
• proof
• Claim 1.2
• proof
• Claim 1.3
• proof
• Claim 1.4
• proof