Optimum-width upward drawings of trees I: Rooted pathwidth

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(logn), while for arbitrary trees it is O(2 √ ). We present algorithms that compute upward drawings with instanceoptimal 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(∆)-approximation where additionally the height is small. In particular any tree has an upward straight-line ordered drawing of area O(∆n logn).