Minimum Rectilinear Polygons for Given Angle Sequences

In this paper, we consider the problem of computing, for a given rectilinear angle sequence, a “small” rectilinear polygon. A rectilinear angle sequence S is a sequence of left (90◦) turns and right (270◦) turns, that is, S = (s1, . . . , sn) ∈ {L, R}, where n is the length of S. As we consider only rectilinear angle sequences, we usually drop the term “rectilinear.” A polygon P realizes an angle sequence S if there is a counterclockwise walk along the interior boundary of P such that the turns at the vertices of P occur in the same order as in S. In order to measure the size of a polygon, we only consider polygons that lie on the integer grid. Then, the area of a polygon P corresponds to the number of grid cells that lie in the interior of P . The bounding box of P is the smallest axis-parallel enclosing rectangle of P . The perimeter of P is the sum of the lengths of the edges of P . The task is, for a given angle sequence S, to find a polygon that realizes S and minimizes (i) (the area of) its bounding box, (ii) its area, or (iii) its perimeter. Fig. 1 shows that, in general, the three criteria cannot be minimized simultaneously. Obviously, the angle sequence of a polygon is unique (up to rotation), but the number of polygons that realize a given angle sequence is unbounded. The formula for the angle sum of a polygon implies that, in any angle sequence, # L = # R+4, where # counts the numbers of L and R symbols. Bae et al. [1] considered, for a given angle sequence S, the polygon P (S) that realizes S and minimizes its area. They studied the following question: Given a length n, find the angle sequence S such that the area of P (S) is minimized (δ(n)) or maximized (∆(n)). They proved that δ(n) ∈ {n/2−1;n/2} and ∆(n) = (n−2)(n+4)/8. In graph drawing, the standard approach to drawing a graph of maximum degree 4 orthogonally (that is, with rectilinear edges) is the topology–shape–metrics approach of Tamassia [4]: (1) Compute a planar(ized) embedding; (2) compute an orthogonal representation, that is, an angle sequence for each edge and an angle for each vertex; (3) compact the graph, that is, draw it inside a bounding box of minimum area. Step (3) has been shown to be NP-complete by Patrignani [3]. Note that an orthogonal representation computed in step (2) is essentially an angle sequence for each face of the planarized embedding, so our problem corresponds to step (3) in the special case that the input graph is a simple cycle.