Tree Drawings with Columns

TL;DR

This paper introduces a novel tree visualization method using columns to encode additional data dimensions, analyzing the complexity of minimizing crossings and providing fixed-parameter tractable algorithms for certain variants.

Contribution

It extends upward tree drawings with columns, explores crossing minimization complexity, and offers FPT algorithms and approximations for these problems.

Findings

Crossing minimization is FPT in maximum degree for strict variants.

NP-hardness results for relaxed variants.

FPT algorithm in crossings plus maximum degree, and an FPT-approximation in degree.

Abstract

Our goal is to visualize an additional data dimension of a tree with multifaceted data through superimposition on vertical strips, which we call columns. Specifically, we extend upward drawings of unordered rooted trees where vertices have assigned heights by mapping each vertex to a column. Under an orthogonal drawing style and with every subtree within a column drawn planar, we consider different natural variants concerning the arrangement of subtrees within a column. We show that minimizing the number of crossings in such a drawing can be achieved in fixed-parameter tractable (FPT) time in the maximum vertex degree $\Delta$ for the most restrictive variant, while becoming NP-hard (even to approximate) already for a slightly relaxed variant. However, we provide an FPT algorithm in the number of crossings plus $\Delta$, and an FPT-approximation algorithm in $\Delta$ via a reduction to feedback arc set.