Optimum-width upward drawings of trees I: Rooted pathwidth.
Thérèse Biedl · arXiv (Cornell University) · 2015
An upward drawing of a rooted tree is a drawing such that no parents are below their children. It is ordered if the edges to children appear in prescribed order around each vertex. It is well-known that any tree has an upward (unordered) drawing with width $\log (n+1)$. For ordered drawings, the best-known bounds for the width for binary trees is $O(\log n)$, while for arbitrary trees it is $O(2^{O(\sqrt{\log n})})$. We present algorithms that compute upward drawings with instance-optimal width, i.e., the width is the minimum-possible for the input tree. In this first paper, we mostly study unordered drawings, where the algorithm is very simple and the drawings obtained are straight-line. We also give 2-approximation algorithms for the width of upward ordered drawings, and $O(\Delta)$-approximation where additionally the height is small. In particular any tree has an upward straight-line ordered drawing of area $O(\Delta n\log n)$.