Minimum-Width Drawing of Trees with Sized Vertices

Abstract

Trees arise in many applications and computing nice tree layouts is a classical problem in information visualization. In many practical settings, vertices need to be represented as rectangles with a given width and height rather than as points. When an order over the children of each vertex is given, polynomial-time algorithms are known that produce drawings adhering to various drawing conventions. However, in many applications, the order of children carries no semantic meaning, and choosing it well can significantly reduce the drawing's width. In this paper, we study the problem \textsc{Min-Width Tree Drawing with Reordering} (\textsc{MWD}): given a rooted tree whose vertices have prescribed widths, find a sibling order at each internal vertex that minimizes the width of the resulting layered drawing. We show that the problem is \textsf{NP}-complete, even on binary trees with unit-width vertices. We present a mixed integer linear program that solves \textsc{MWD} exactly on moderately sized instances, and a heuristic that is fast and delivers good results in practice. We evaluate both approaches against a baseline on synthetic and real-world datasets, where reordering reduces drawing width by a median of $\approx20\%$ and by up to $\approx55\%$ on individual instances. The heuristic computes its layouts in under a second and, when the MILP proves optimality, it stays within $25\%$ of the optimal width in three-quarters of all instances.