We make progress on a number of open problems concerning the area requirement for drawing trees on a grid. We prove that
Let T be an n-node tree of maximum degree 4, and let P be a set of n points in the plane with no two points on the same horizontal or vertical line. It is an open question whether T always has a planar drawing on P such that each edge is drawn as an orthogonal path with one bend (an "L-shaped" edge). By giving new methods for drawing trees, we improve the bounds on the size of the point set P for which such drawings are possible to: O(n^{1.55}) for maximum degree 4 trees; O(n^{1.22}) for maximum degree 3 (binary) trees; and O(n^{1.142}) for perfect binary trees.
Drawing ordered trees with L-shaped edges is harder---we give an example that cannot be done and a bound of O(n log n) points for L-shaped drawings of ordered caterpillars, which contrasts with the known linear bound for unordered caterpillars.
We investigate the problem of constructing planar drawings with few bends for two related problems, the partially embedded graph (PEG) problem to extend a straight-line planar drawing of a subgraph to a planar drawing of the whole graph and the simultaneous planarity (SEFE) problem to find planar drawings of two graphs that coincide on shared vertices and edges. In both cases we show that if the required planar drawings exist, then there are planar drawings with a linear number of bends per edge and, in the case of simultaneous planarity, a constant number of crossings between every pair of edges. Our proofs provide efficient algorithms if the combinatorial embedding information about the drawing is given. Our result on partially embedded graph drawing generalizes a classic result of Pach and Wenger showing that any planar graph can be drawn with fixed locations for its vertices and with a linear number of bends per edge.
We consider the problem of finding a planar embedding of a graph at fixed vertex locations that minimizes the total edge length. The problem is known to be NP-hard. We give polynomial time algorithms achieving an O(n^{1/2} log n) approximation for paths and matchings, and an O(n) approximation for general graphs.
In 1944, Cairns proved the following theorem: given any two straight-line planar drawings of a triangulation with the same outer face, there exists a morph (i.e., a continuous transformation) between the two drawings so that the drawing remains straight-line planar at all times. Cairns's original proof required exponentially many morphing steps. We prove that there is a morph that consists of O(n^2) steps, where each step is a linear morph that moves each vertex at constant speed along a straight line. Using a known result on compatible triangulations this implies that for a general planar graph G and any two straight-line planar drawings of G with the same embedding, there is a morph between the two drawings that preserves straight-line planarity and consists of O(n^4) steps.
In this paper we introduce self-approaching graph drawings. A straight-line drawing of a graph is self-approaching if, for any origin vertex s and any destination vertex t, there is an st-path in the graph such that, for any point q on the path, as a point p moves continuously along the path from the origin to q, the Euclidean distance from p to q is always decreasing. This is a more stringent condition than a greedy drawing (where only the distance between vertices on the path and the destination vertex must decrease), and guarantees that the drawing is a 5.33-spanner.
We study three topics: (1) recognizing self-approaching drawings; (2) constructing self-approaching drawings of a given graph; (3) constructing a self-approaching Steiner network connecting a given set of points.
We show that: (1) there are efficient algorithms to test if a polygonal path is self-approaching in R^2 and R^3, but it is NP-hard to test if a given graph drawing in R^3 has a self-approaching uv-path; (2) we can characterize the trees that have self-approaching drawings; (3) for any given set of terminal points in the plane, we can find a linear sized network that has a self-approaching path between any ordered pair of terminals.
We give a tradeoff theorem between the area and the aspect ratio required by any planar straight-line drawing of K_{2,n} on the integer lattice. In particular we show that if the drawing is contained in a rectangle of area O(n) then the rectangle must have aspect ratio Omega(n), and conversely, if the aspect ratio is 1 then the area must be Omega(n^2/log^2 n).
We investigate the problem of drawing an arbitrary n-node binary tree orthogonally and upwardly in an integer grid using straight-line edges. We show that one can simultaneously achieve good area bounds while also allowing the aspect ratio to be chosen as being a constant or even an arbitrary parameter. In addition, we show that one can also achieve an additional desirable aesthetic criterion, which we call "subtree separation." Our drawings require O(n log n) area, which is optimal to within a constant factor. An improvement for non-upward drawings is briefly mentioned.
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