Upward Octagonal Drawings of Ternary Trees

Abstract

We explore ways to embed a ternary tree in an integer coordinate grid such that the width of the drawing is minimized. We provide upper and lower bounds on the width requirement of planar, straight-line, upward, order-preserving drawings of ternary trees in an octagonal grid. We present a linear-time algorithm for constructing such octagonal grid drawings of any $n$-node ternary tree with $O(n^{0.68})$ width. (This bound can be improved to $O(n^{0.631})$ width in the so-called HVA-model.) For ideal octagonal grid drawings of complete $n$-node ternary trees, we provide an $\Omega(n^{0.411})$ width lower bound.