Minimum rectilinear polygons for given angle sequences

Abstract

A rectilinear polygon is a simple polygon whose edges are axis-aligned. Walking counterclockwise on the boundary of such a polygon yields a sequence of left turns and right turns. The number of left turns always equals the number of right turns plus four. It is known that any such sequence can be realized by a rectilinear polygon.

In this paper, we consider the problem of finding realizations that minimize the perimeter or the area of the polygon or the area of the bounding box of the polygon. We show that all three problems are $\mathsf{NP}$ -hard in general. This answers an open question of Patrignani (2001) [13], who showed that it is $\mathsf{NP}$ -hard to minimize the area of the bounding box of an orthogonal drawing of a given planar graph. We also show that realizing a polyline within a bounding box of minimum area (or within a fixed given rectangle) is $\mathsf{NP}$ -hard. Then we consider the special cases of x-monotone and xy-monotone rectilinear polygons. For these, we can optimize the three objectives efficiently.

Introduction

In this paper, we consider the problem of computing, for a given rectilinear angle sequence, a “small” rectilinear polygon that realizes the sequence. A rectilinear angle sequence S is a sequence of left ($+{90}^{\circ }$) turns, denoted by L, and right ($-{90}^{\circ }$) turns, denoted by R. We write $S=(s_1,\dots ,s_n)\in {\{\mathtt{\text{L}},\mathtt{\text{R}}\}}^n$, where n is the length of S. As we consider only rectilinear angle sequences, we usually drop the term “rectilinear.” A rectilinear polygon P realizes an angle sequence S if there is a counterclockwise (ccw) walk along the boundary of P such that the turns at the vertices of P, encountered during the walk, form the sequence S. The turn at a vertex v of P is a left or right turn if the interior angle at v is ${90}^{\circ }$ (v is convex) or, respectively, ${270}^{\circ }$ (v is reflex). We call the problem Minimum Rectilinear Polygon for Given Angle Sequence.

In order to measure the size of a polygon, we only consider polygons that lie on the integer grid. In this context, 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 simple⁴ polygon that realizes S and minimizes

• its bounding box (area),

• its area, or

• 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, $n=2r+4$, where n is the length of the sequence and r is the number of right turns [17]. In other words, the number of right turns is exactly four less than the number of left turns.

Related work  Bae et al. [2] considered, for a given angle sequence S, the polygon $P(S)$ that realizes S and minimizes its area. Given a number n, let $\delta (n)$ be the smallest area of $P(S)$ over all angle sequences S of length n, and let $\mathrm{\Delta }(n)$ be the largest area of $P(S)$ over all angle sequences S of length n. They showed

• $\delta (n)=n/2-1$ if $n\equiv 4\mathrm{mod}8$, $\delta (n)=n/2$ otherwise, and

• $\mathrm{\Delta }(n)=(n-2)(n+4)/8$ for any n with $n\ge 4$.

The result for $\mathrm{\Delta }(n)$ tells us that any angle sequence S of length n can be realized by a polygon with area at most $(n-2)(n+4)/8$.

Several authors have explored the problem of realizing a turn sequence. Culberson and Rawlins [6] and Hartley [11] described algorithms that, given a sequence of exterior angles summing up to 2π, construct a simple polygon realizing that angle sequence. Culberson and Rawlins' algorithm, when constrained to $\pm {90}^{\circ }$ angles, produces polygons with no colinear edges, implying that any n-vertex polygon can be drawn with area approximately ${(n/2-1)}^2$. However, as Bae et al. [2] showed, the bound is not tight. In his PhD thesis, Sack [15] introduced label sequences (which are equivalent to turn sequences) and, among others, developed a grammar for label sequences that can be realized as simple rectilinear polygons. Vijayan and Wigderson [17] considered the problem of drawing rectilinear graphs, of which rectilinear polygons are a special case, using an edge labeling that is equivalent to a turn sequence in the case of paths and cycles.

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 [16]:

• Compute a planar(ized) embedding, that is, a circular order of edges around each vertex that admits a crossing-free drawing;

• compute an orthogonal representation, that is, an angle sequence for each edge and an angle for each vertex;

• compact the graph, that is, draw it inside a bounding box of minimum area.

Step (3) is NP-hard⁵ for planar graphs as shown by Patrignani [13]. For non-planar graphs, it is even inapproximable within a polynomial factor unless $\mathsf{P}=\mathsf{NP}$ as shown by Bannister et al. [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.

Another related problem is to reconstruct a simple (non-rectilinear) polygon from partial geometric information. Disser et al. [7] constructed a simple polygon in $\mathcal{O}(n^3\log n)$ time from an ordered sequence of angles measured at the vertices visible from each vertex. Chen and Wang [5] showed how to solve the problem in $\mathcal{O}(n^2)$ time, which is optimal in the worst-case. Biedl et al. [4] considered polygon reconstruction from points (instead of angles) captured by laser scanning devices. Very recently, Asaeedi et al. [1] showed how to enclose a given set of points by a simple polygon whose vertices are a subset of the points and that optimizes some criteria (minimum area, maximum perimeter or maximum number of vertices). The vertex angles are constrained to lie below a threshold.

Our contribution  We show that finding a minimum polygon that realizes a given angle sequence is $\mathsf{NP}$ -hard for any of the three measures: area of the bounding box, polygon area, and polygon perimeter; see Section 2. This hardness result extends the one of Patrignani [13] and settles an open question that he posed. We note that in an extended abstract [8] of this paper there were some inaccuracies in our proof that now have been addressed. As a warm-up, we show that realizing an angle sequence as a polyline within a given rectangle is $\mathsf{NP}$ -hard. We then infer that it is also $\mathsf{NP}$ -hard to find a realizing polyline within a bounding box of minimum area.

We also give efficient algorithms for special types of angle sequences, namely xy- and x-monotone sequences, which are realized by xy-monotone and x-monotone polygons, respectively. For example, Fig. 1 shows that LLRRLLRLLRLRLLRLRLLR is an x-monotone sequence. Our algorithms for these angle sequences minimize the bounding box and the area (Section 3) and the perimeter (Section 4). For an overview of our results, see Table 1. Throughout this paper, a segment is always an axis-aligned line segment.