Orthogonal Polygons Research Articles

Area-universal drawings of biconnected outerplane graphs

Contact graph representation is a classical graph drawing style where vertices are represented by geometric objects such that edges correspond to contacts between the objects. Contact graph representations using axis-aligned rectilinear polygons are well-investigated. On the other hand, only a scarcity of results and techniques are available for cases using polygons that are not necessarily rectilinear. In this paper, we investigate a type of contact graph representations (named t-TkR) using k-sided convex polygons with their boundaries being t-sided. Given a biconnected outerplane graph, we present a clean necessary and sufficient condition for the graph to admit a t-TkR. We give a linear time algorithm for constructing an area-universal 3-T4R of a given biconnected outerplane graph, which is of interest since most of the previous results on area-universal drawings are with respect to rectilinear settings.

Partitioning orthogonal polygons into ≤ 8-vertex pieces, with application to an art gallery theorem

We prove that every simply connected orthogonal polygon of n vertices can be partitioned into ⌊3n+416⌋ (simply connected) orthogonal polygons of at most 8 vertices. It yields a new and shorter proof of the theorem of A. Aggarwal that ⌊3n+416⌋ mobile guards are sufficient to control the interior of an n-vertex orthogonal polygon. Moreover, we strengthen this result by requiring combinatorial guards (visibility is only needed at the endpoints of patrols) and prohibiting intersecting patrols. This yields positive answers to two questions of O'Rourke [7, Section 3.4]. Our result is also a further example of the “metatheorem” that (orthogonal) art gallery theorems are based on partition theorems.

Specifying the staircase kernel of a two-fold connected orthogonal polygon

Let $$S\ne \phi $$ be a two-fold connected orthogonal polygon in the plane. Assume that S is starshaped via staircase paths and K is any component of $$\textit{Ker} S$$ , the staircase kernel of S. Let B be the bounded component of $$ \mathbb {R}^2{\setminus } S$$ . If B contains one kind of north, east, south and west dents then $$\textit{Ker} S$$ is either one component or two and the positions of these components agree with the kind of dents. But if B contains two kinds of north, east, south and west dents then either KerS is one component or S is not starshaped via staircase paths.

On The Illumination of Polygons by 60 -Floodlights

In this paper, first we consider the problem of finding the minimum number of floodlights that can illuminate the interior of an orthogonal polygon when the range of illumination is restricted to 60°. Then we study this problem for the pseudo-triangles. In general, we want to guard an environment with cameras in order to ensure that every point in the environment is seen from at least one camera. We intend to minimize the total number of cameras required and also the cameras can see a limited range. This problem is answered before for some range of visions but there was only a conjecture for 60°-cameras [6]. We prove the correctness of this conjecture for the orthogonal polygons and the pseudo-triangles.

An Effective Chemical Mechanical Polishing Fill Insertion Approach

To reduce chip-scale topography variation, dummy fill is commonly used to improve the layout density uniformity. Previous works either sought the most uniform density distribution or sought to minimize the inserted dummy fills while satisfying certain density uniformity constraint. However, due to more stringent manufacturing challenges, more criteria, like line deviation and outlier, emerge at newer technology nodes. This article presents a joint optimization scheme to consider variation, total fill, line deviation, outlier, overlap, and running time simultaneously. More specifically, first we decompose the rectilinear polygons and partition fillable regions into rectangles for easier processing. After decomposition, we insert dummy fills into the fillable rectangular regions optimizing the fill metrics simultaneously. We propose three approaches, Fast Median approach, LP approach, and Iterative approach, which are much faster with better quality, compared with the results of the top three contestants in the ICCAD Contest 2014.

Evaluation of Labeling Strategies for Rotating Maps

We consider the following problem of labeling points in a dynamic map that allows rotation. We are given a set of feature points in the plane labeled by a set of mutually disjoint labels, where each label is an axis-aligned rectangle attached with one corner to its respective point. We require that each label remains horizontally aligned during the map rotation, and our goal is to find a set of mutually nonoverlapping visible labels for every rotation angle α ∈ [0, 2π) so that the number of visible labels over a full map rotation of 2π is maximized. We discuss and experimentally evaluate several labeling strategies that define additional consistency constraints on label visibility to reduce flickering effects during monotone map rotation. We introduce three heuristic algorithms and compare them experimentally to an existing approximation algorithm and exact solutions obtained from an integer linear program. Our results show that on the one hand, low flickering can be achieved at the expense of only a small reduction in the objective value, and on the other hand, the proposed heuristics achieve a high labeling quality significantly faster than the other methods.

On the number of anchored rectangle packings for a planar point set

We consider packing axis-aligned rectangles r1,…,rn in the unit square [0,1]2 such that a vertex of each rectangle ri is a given point pi (i.e., ri is anchored at pi). We explore the combinatorial structure of all locally maximal configurations. When the given points are the lower-left corners of the rectangles, the number of maximal packings is shown to be at most 2nCn, where Cn is the nth Catalan number. The number of maximal packings remains exponential in n when the points may be arbitrary corners of the rectangles. Both upper bounds are complemented with exponential lower bounds. Finally, we define the graph of all lower-left anchored maximal rectangle packings, where the edges correspond to elementary operations between two packings, which leads to an output sensitive algorithm for computing these packings.

Improved Approximation Algorithms for Box Contact Representations

We study the following geometric representation problem: Given a graph whose vertices correspond to axis-aligned rectangles with fixed dimensions, arrange the rectangles without overlaps in the plane such that two rectangles touch if the graph contains an edge between them. This problem is called \textsc{Contact Representation of Word Networks} (\textsc{Crown}) since it formalizes the geometric problem behind drawing word clouds in which semantically related words are close to each other. \textsc{Crown} is known to be NP-hard, and there are approximation algorithms for certain graph classes for the optimization version, \textsc{Max-Crown}, in which realizing each desired adjacency yields a certain profit. We present the first $O(1)$-approximation algorithm for the general case, when the input is a complete weighted graph, and for the bipartite case. Since the subgraph of realized adjacencies is necessarily planar, we also consider several planar graph classes (namely stars, trees, outerplanar, and planar graphs), improving upon the known results. For some graph classes, we also describe improvements in the unweighted case, where each adjacency yields the same profit. Finally, we show that the problem is APX-hard on bipartite graphs of bounded maximum degree.

How to Guard Orthogonal Polygons: Diagonal Graphs and Vertex Covers

We extend and unify most known results about guarding orthogonal polygons by introducing the same-sign diagonal graphs of a convex quadrangulation and applying results about vertex covers for graphs. Our approach also yields new theorems and often guarantees two disjoint vertex guard sets of relatively small cardinality. For instance, an orthogonal polygon on n vertices has two disjoint vertex guard sets of cardinality at most $$\lfloor n/4\rfloor $$?n/4?. We give new proofs of Aggarwal's one-hole theorem and the orthogonal fortress theorem. We prove that an orthogonal polygon with n vertices and any number of holes can be protected by at most $$\lfloor (17n-8)/52\rfloor $$?(17n-8)/52? vertex guards, improving the best known bound of $$\lfloor n/3\rfloor $$?n/3?. Also, an orthogonal polygon with n vertices and h holes can be protected by $$\lfloor (n+2h)/3\rfloor $$?(n+2h)/3? guarded guards, which is best possible when $$n\ge 16h$$n?16h. Moreover, for orthogonal fortresses with n vertices, $$\lfloor (n+6)/3\rfloor $$?(n+6)/3? guarded guards are always sufficient and sometimes necessary.

Partitions of Rectilinear Polygons with Minimum Stabbing Number

We study integer programming models for the problem of finding a rectangular partition of a rectilinear polygon with minimum stabbing number. Polyhedral investigations and computational tests are reported for the first time for the problem.

An Upper Bound on the k-Modem Illumination Problem

A variation on the classical polygon illumination problem was introduced in [Aichholzer et al. EuroCG’09]. In this variant light sources are replaced by wireless devices called [Formula: see text]-modems, which can penetrate a fixed number [Formula: see text], of “walls”. A point in the interior of a polygon is “illuminated” by a [Formula: see text]-modem if the line segment joining them intersects at most [Formula: see text] edges of the polygon. It is easy to construct polygons of [Formula: see text] vertices where the number of [Formula: see text]-modems required to illuminate all interior points is [Formula: see text]. However, no non-trivial upper bound is known. In this paper we prove that the number of kmodems required to illuminate any polygon of [Formula: see text] vertices is [Formula: see text]. For the cases of illuminating an orthogonal polygon or a set of disjoint orthogonal segments, we give a tighter bound of [Formula: see text]. Moreover, we present an [Formula: see text] time algorithm to achieve this bound.

Bivariate shape preserving interpolation: A fractal-classical hybrid approach

The notion of cubic fractal interpolation function (FIF) has received considerable attention in the literature due to its versatility, flexibility and ease of implementation. In this article, we shall view cubic FIFs as a family of C1-continuous fractal functions associated with the traditional C1-continuous cubic spline. General theorems that identify suitable values of the parameters so as to constrain a fractal function and its first derivative within suitable axis-aligned rectangles are reported. By applying these theorems, cubic fractal interpolation of a data set subject to strip conditions on the interpolant and its first derivative is discussed. These results are applied to investigate positivity and monotonicity properties of a hybrid bivariate interpolant over a rectangular region Robtained by blending univariate cubic FIFs via bicubically blended Coons patch. The L∞-norm of the error in approximating a function f ∈ C2(R)with the proposed bivariate interpolant is shown to be of order O(h2) as h → 0.

Computer-Aided Design of Layouts for Cutting Shells for Chemical and Oil-and-Gas Engineering Articles

The results of theoretical and experimental studies of new models and methods for creating software for an automated (computer-based) information system to solve direct and inverse problems of optimal geometric cover of a multiply connected orthogonal polygon (shell) of chemical and oil-and-gas engineering articles are analyzed.

Ramified Rectilinear Polygons: Coordinatization by Dendrons

Simple rectilinear polygons (i.e. rectilinear polygons without holes or cutpoints) can be regarded as finite rectangular cell complexes coordinatized by two finite dendrons. The intrinsic $l_1$-metric is thus inherited from the product of the two finite dendrons via an isometric embedding. The rectangular cell complexes that share this same embedding property are called ramified rectilinear polygons. The links of vertices in these cell complexes may be arbitrary bipartite graphs, in contrast to simple rectilinear polygons where the links of points are either 4-cycles or paths of length at most 3. Ramified rectilinear polygons are particular instances of rectangular complexes obtained from cube-free median graphs, or equivalently simply connected rectangular complexes with triangle-free links. The underlying graphs of finite ramified rectilinear polygons can be recognized among graphs in linear time by a Lexicographic Breadth-First-Search. Whereas the symmetry of a simple rectilinear polygon is very restricted (with automorphism group being a subgroup of the dihedral group $D_4$), ramified rectilinear polygons are universal: every finite group is the automorphism group of some ramified rectilinear polygon.

An improved star test for implicit polynomial objects

For a given point set, a particular point is called a star if it can see all the boundary points of the set. The star test determines whether a candidate point is a star for a given set. It is a key component of some topology computing algorithms such as Connected components via Interval Analysis (CIA), Homotopy type via Interval Analysis (HIA), etc. Those algorithms decompose the input object using axis-aligned boxes, so that each box is either not intersecting or intersecting with the object and in this later case its center is a star point of the intersection. Graphs or simplicial complexes describing the topology of the objects can be obtained by connecting these star points following different rules. The star test is performed for simple primitive geometric objects, because complex objects can be constructed using Constructive Solid Geometry (CSG), and the star property is preserved via union and intersection. In this paper, we improve the method to perform the test for implicit objects. For a primitive set defined by an implicit polynomial equation, the polynomial is made homogeneous with the introduction of an auxiliary variable, thus the degree of the star condition is reduced. A linear programming optimization is introduced to further improve the performance. Several examples are given to show the experimental results of our method.

Automatic Generation of 3D Building Models for Sustainable Development

3D building models are important in several fields, such as urban planning and BIM (Building Information Model). However, enormous time and labor are required to create these 3D models. In order to automate laborious steps, a GIS and CG integrated system have been proposed for the automatic generation of 3D building models, based on building polygons (building footprints) on digital maps. These polygons are usually orthogonal. A complicated orthogonal polygon can be partitioned into a set of rectangles. The proposed integrated system partitions and rectifies orthogonal building polygons into a set of rectangles and places rectangular roofs and box-shaped building bodies on these rectangles. In this paper, in addition, for greater solar photovoltaic (PV) generation, a double leaned roof attached by PV arrays is created. The sizes and positions, slopes of roof boards, and under roof constructions are clarified by designing the top view and side view of a double leaned roof.

Fitting Planar Graphs on Planar Maps

Graph and cartographic visualization have the common objective to provide intuitive understanding of some underlying data. We consider a problem that combines aspects of both by studying the problem of fitting planar graphs on planar maps. After providing an NP-hardness result for the general decision problem, we identify sufficient conditions so that a fit is possible on a map with rectangular regions. We generalize our techniques to non-convex rectilinear polygons, where we also address the problem of efficient distribution of the vertices inside the map regions.

A rational iterated function system for resolution of univariate constrained interpolation

Iterated Function Systems (IFSs) provide a standard framework for generating Fractal Interpolation Functions (FIFs) that yield smooth or non-smooth approximants. Nevertheless, the most widely studied FIFs so far in the literature that are obtained through polynomial IFSs are, in general, incapable of reproducing important shape properties inherent in a given data set. Abandoning the polynomiality of the functions defining the IFS, we introduce a new class of rational IFS that generates fractal functions (self-referential functions) for solving constrained interpolation problems. Suitable values of the rational IFS parameters are identified so that: (i) the corresponding FIF inherits positivity and/or monotonicity properties present in the data set, and (ii) the attractor of the IFS lies within an axis-aligned rectangle. The proposed IFS schemes for the shape preserving interpolation generalize some of the classical non-recursive interpolation methods, and expand the interpolation/approximation, including approximants for which functions themselves or the first derivatives can even be non-differentiable in a dense set of points of the domain. For appropriate values of the IFS parameters, the resulting rational quadratic FIF converges uniformly to the original function $$\varPhi \in \mathcal {C}^3[x_1, x_n]$$ with $$h^3$$ order of convergence, where $$h$$ denotes the norm of the partition. We also provide a number of examples intended to demonstrate the proposed schemes and to suggest how these schemes outperform their classical counterparts.

Algorithm for the Removal of Rectangle Containment for Rectangle Spline Generation

ABSTRACTGiven a set of axis-aligned rectangles in the plane, this paper presents an algorithm for the removal of rectangle containment as well as rectangle enclosures. That is, given a query rectangle Rq, this algorithm removes all rectangles Rk, … , Rl from the rectangle set that are contained in Rq. It also removes Rq if there is a rectangle Rj that encloses Rq. The algorithm, initially implemented for rectangle spline generation using rectangles as building blocks, is fast, easy to implement and maintain, it can be run on parallel machines such as GPUs at no extra cost, but it requires extra memory to store the rectangle indexes for quick access and space partitioning.

Cut equivalence of [formula omitted]-dimensional guillotine partitions

A guillotine partition of a d-dimensional axis-aligned box B is a recursive partition of B by axis-aligned hyperplane cuts. The size of a guillotine partition is the number of boxes it contains. Two guillotine partitions are box-equivalent if their boxes satisfy compatible order relations with respect to the axes. (In many works, box-equivalent guillotine partitions are considered identical.) In the present work we define cut-equivalence of guillotine partitions, derived in a similar way from order relations of cuts. We prove structural properties related to these kinds of equivalence, and enumerate cut-equivalence classes of d-dimensional guillotine partitions of size n.