Optimal Monotone Drawings of Trees

A monotone drawing of a graph $G$ is a straight-line drawing of $G$ such that, for every pair of vertices $u,w$ in $G$, there exists a path $P_{uw}$ in $G$ that is monotone in some direction $l_{uw}$. (Namely, the order of the orthogonal projections of the vertices of $P_{uw}$ on $l_{uw}$ is the same as the order in which they appear in $P_{uw}$.) The problem of finding monotone drawings for trees has been studied in several recent papers. The main focus is to reduce the size of the drawing. Currently, the smallest drawing size is $O(n\log n) \times O(n\log n)$. In this paper, we present an algorithm for constructing monotone drawings of trees on a grid of size at most $12n \times 12n$. The smaller drawing size is achieved by a procedure that carefully assigns primitive vectors to the paths of the input tree $T$. We also show that there exists a tree $T_0$ such that any monotone drawing of $T_0$ must use a grid of size at least $n/9 \times n/9$. So the size of our monotone drawings is asymptotically optimal.