Area-efficient algorithms for straight-line tree drawings

Abstract

We investigate several straight-line drawing problems for bounded-degree trees in the integer grid without edge crossings under various types of drawings: (1) upward drawings whose edges are drawn as vertically monotone chains, a sequence of line segments, from a parent to its children, (2) order-preserving drawings which preserve the left-to-right order of the children of each vertex, and (3) orthogonal straight-line drawings in which each edge is represented as a single vertical or horizontal segment. Main contribution of this paper is a unified framework to reduce the upper bound on area for the straight-line drawing problems from O(n log n) (Crescenzi et al., 1992) to O(n log log n). This is the first solution of an open problem stated by Garg et al. (1993). We also show that any binary tree admits a small area drawing satisfying any given aspect ratio in the orthogonal straight-line drawing type. Our results are briefly summarized as follows. Let T be a bounded-degree tree with n vertices. Firstly, we show that T admits an upward straight-line drawing with area O(n log log n). If T is binary, we can obtain an O(n log log n)-area upward orthogonal drawing in which each edge is drawn as a chain of at most two orthogonal segments and which has O(n/log n) bends in total. Secondly, we present O(n log log n)-area (respectively, -volume) orthogonal straight-line drawing algorithms for binary trees with arbitrary aspect ratios in 2-dimension (respectively, 3-dimension). Finally, we present some experimental results which shows the area requirements, in practice, for (order-preserving) upward drawing are much smaller than theoretical bounds obtained through analysis. (C) 2000 Elsevier Science B.V. All rights reserved.