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Minimum-Width Drawing of Trees with Sized Vertices

Authors Markus Wallinger , Oscar Navarro , Stephen G. Kobourov

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LIPIcs.GD.2026.7.pdf
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Subject Classification

ACM Subject Classification
  • Human-centered computing → Graph drawings
  • Theory of computation → Computational geometry
Keywords
  • Tree drawing
  • sized vertices
  • minimum width
  • integer linear programming
  • heuristics

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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 Min-Width Tree Drawing with Reordering (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 NP-complete, even on binary trees with unit-width vertices. We present a mixed integer linear program that solves 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 ≈20% and by up to ≈55% 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.

Cite As Get BibTex

Markus Wallinger, Oscar Navarro, and Stephen G. Kobourov. Minimum-Width Drawing of Trees with Sized Vertices. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 7:1-7:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.GD.2026.7

Author Details

Markus Wallinger
  • Technical University of Munich, Germany
Oscar Navarro
  • Technical University of Munich, Germany
Stephen G. Kobourov
  • Technical University of Munich, Germany

Supplementary Materials

References

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