Parallel h-v Drawings of Binary Trees

1 I n t r o d u c t i o n In this paper we examine drawings of rooted binary trees. We study the h-v drawing convention studied by Crescenzi, Di Bat t is ta and Piperno [3] and Eades, Lin and Lin [7]. Our results extend to the inclusion convention [6], and to slicing floorplanning [10, 2]. The drawing of a rooted binary tree using the h-v drawing convention is a planar grid drawing in which tree nodes are represented as points (of integer coordinates) in the plane and tree edges as non-overlapping vertical or horizontal line segments. Moreover, each node is placed immediately to the right or immediately below its parent and the drawings of subtrees rooted at nodes with the same parent are non-overlapping. Figure 1 shows three different h-v drawings of the same tree. Different h-v drawings of the same tree can be of different quality. The quality (or cost) is a function of the drawing. The most commonly used cost function is the area of the enclosing rectangle of the drawing. Eades et al. [7] showed how to compute in O(n 2) t ime an optimal h-v drawing of a tree with n nodes with respect to a cost function r h) which is nondecreasing in both parameters w and h, where w and h are the width and the height of the enclosing rectangle of the drawing, respectively. The same method can be used to develop optimal drawings (with respect to some cost function r when the inclusion convention is adopted. In the inclusion convention, a node is represented by a rectangle and the parent-child relation by enclosing the rectangle which represents the child within that of the parent. Moreover, rectangles of * The work of the second author is partially supported by the EEC ESPRIT Basic Research Action No. 7141 (ALCOM II) and by the NSF grant No. CDA-9211155. Emaih [email protected], [email protected], symvonis~cs.su.oz.au