Area-efficient algorithms for upward straight-line tree drawings

Abstract

In this paper, we investigate planar upward straight-line grid drawing problems for bounded-degree rooted trees so that a drawing takes up as little area as possible. A planar upward straight-line grid tree drawing satisfies the following four constraints: (1) all vertices are placed at distinct grid points (grid), (2) all edges are drawn as straight lines (straight-line), (3) no two edges in the drawing intersect (planar), and (4) no parents are placed below their children (upward). Our results are summarized as follows. First, we show that a bounded-degree tree T with n vertices 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. Second, we show that bounded-degree trees in some classes of balanced trees, frequently used as search trees, admit strictly upward straight-line drawings with area O(n loglog n). They include k-balanced trees, red-black trees, BB [α]-trees, and (a, b)-trees. In addition, trees in the same classes admit O(n(loglog n)2)-area strictly upward straight-line drawings that preserve the left-to-right ordering of the children of each vertex. Finally, we discuss an extension of our drawing algorithms to non-upward straight-line drawing algorithms in 2- and 3-dimensions.