Orthogonal Polygons Research Articles

Covering rectilinear polygons by rectangles

Three approximation algorithms to cover a rectilinear polygon that is neither horizontally nor vertically convex by rectangles are developed. All three guarantee covers that have at most twice as many rectangles as in an optimal cover. One takes O(n log n) time, where n is the number of vertices in the rectilinear polygon. The other two take O(n/sup 2/) and O(n/sup 4/) time. Experimental results indicate that the algorithms with complexity O(n/sup 2/) and O(n/sup 4/) often obtain optimal or near-optimal covers. >

Covering orthogonal polygons with star polygons: The perfect graph approach

This paper studies the combinatorial structure of visibility in orthogonal polygons. We show that the visibility graph for the problem of minimally covering simple orthogonal polygons with star polygons is perfect. A star polygon contains a point p, such that for every point q in the star polygon, there is an orthogonally convex polygon containing p and q. This perfectness property implies a polynomial algorithm for the above polygon covering problem. It further provides us with an interesting duality relationship. We first establish that a minimum clique cover of the visibility graph of a simple orthogonal polygon corresponds exactly to a minimum star cover of the polygon. In general, simple orthogonal polygons can have concavities (dents) with four possible orientations. In this case, we show that the visibility graph is weakly triangulated. We thus obtain an O(n 8) algorithm. Since weakly triangulated graphs are perfect, we also obtain an interesting duality relationship. In the case where the polygon has at most three dent orientations, we show that the visibility graph is triangulated or chordal. This gives us an O(n 3) algorithm.

The Boolean Basis Problem and How to Cover Some Polygons by Rectangles

For $S \subseteq \{ 0,1\} ^n $ the Boolean basis (or set basis) problem is to find a minimum size set $B \subseteq \{ 0,1\} ^n $ such that each $s \in S$ is a Boolean sum of vectors from B. This paper examines tractable special cases of this NP-complete problem. In particular a construction preserving tractability is given. The rectangle cover problem—expressing a rectilinear polygon as the union of a minimum number of rectangles—is a main application.

Orthogonally convex coverings of orthogonal polygons without holes

The problem of covering a polygon with convex polygons has proven to be very difficult, even when restricted to the class of orthogonal polygons using orthogonally convex covers. We develop a method of analysis based on dent diagrams for orthogonal polygons and are able to show that Keil's O( n 2) algorithm for covering horizontally convex polygons is asymptotically optimal, but it can be improved to O( n) for counting the number of polygons required for a minimal cover. We also give an optimal O( n 2) covering algoring and an O( n ) counting algorithm for another subclass of orthogonal polygons. Finally, we develop a method of signatures which can be used to obtain polynomial time algorithms for an even larger class of orthogonal polygons.

Perfect Graphs and Orthogonally Convex Covers

We consider the problem of covering simple orthogonal polygons with convex orthogonal polygons. In the case of horizontally or vertically convex polygons we show that the polygon covering problem can be reduced to the problem of covering a permutation graph with minimum number of cliques. In general, orthogonal polygons can have concavities (dents) with four possible orientations. In the case where the polygon has three dent orientations, we show that the polygon covering problem can be reduced to the problem of covering a weakly triangulated graph with a minimum number of cliques. Since weakly triangulated graphs are perfect, we obtain the following duality relationship: the minimum number of orthogonally convex polygons needed to cover an orthogonal polygon P with at most three dent orientations is equal to the maximum number of points of P, no two of which can be contained together in an orthogonally convex covering polygon. Finally, we show that in th case of orthogonal polygons with all four dent orientations, the above duality relationship fails to hold.

Performance Guarantees on a Sweep-Line Heuristic for Covering Rectilinear Polygons with Rectangles

Finding the minimum number of rectangles required to cover a rectilinear or orthogonal polygon, where overlapping of rectangles is allowed, is one of several well-known, hard geometric decomposition problems. This paper reports the first results known that give worst-case performance bounds for an approximation algorithm for this problem. It is proved that partitioning the polygon into rectangles (with no overlapping) produces at most $2\theta + h - 1$ rectangles, where $\theta $ is the minimum number of rectangles in a cover, and h is the number of holes. Examples are also given in which this bound is tight. The proof is based on counting arguments, and on Euler’s formula for a planar graph. This paper shows that extending rectangles vertically and deleting duplicates produces at most $O(\theta \log \theta )$ rectangles. For the proof, geometric constraints are used to construct a large an-tirectangle, a set of points with no two contained in the same rectangle. This algorithm has an $O(n\log n)$ impleme...

Mesh computer algorithms for computational geometry

Asymptotically optimal parallel algorithms are presented for use on a mesh computer to determine several fundamental geometric properties of figures. For example, given multiple figures represented by the Cartesian coordinates of n or fewer planar vertices, distributed one point per processor on a two-dimensional mesh computer with n simple processing elements, Theta (n/sup 1/2/>or=-time algorithms are given for identifying the convex hull and smallest enclosing box of each figure. Given two such figures, a Theta (n/sup 1/2/>or=-time algorithm is given to decide if the two figures are linearly separable. Given n or fewer planar points, Theta (n/sup 1/2/>or=-time algorithms are given to solve the all-nearest neighbor problems for points and for sets of points. Given n or fewer circles, convex figures, hyperplanes, simple polygons, orthogonal polygons, or iso-oriented rectangles, Theta (n/sup 1/2/>or=-time algorithms are given to solve a variety of area and intersection problems. Since any serial computer has worst-case time of Omega (n) when processing n points, these algorithms show that the mesh computer provides significantly better solutions to these problems. >

The orthogonal convex skull problem

We give a combinatorial definition of the notion of a simple orthogonal polygon beingk-concave, wherek is a nonnegative integer. (A polygon is orthogonal if its edges are only horizontal or vertical.) Under this definition an orthogonal polygon which is 0-concave is convex, that is, it is a rectangle, and one that is 1-concave is orthoconvex in the usual sense, and vice versa. Then we consider the problem of computing an orthoconvex orthogonal polygon of maximal area contained in a simple orthogonal polygon. This is the orthogonal version of the potato peeling problem. AnO(n2) algorithm is presented, which is a substantial improvement over theO(n7) time algorithm for the general problem.

Optimum watchman routes

In this paper we consider the problem of finding shortest routes from which every point in a given space is visible (watchman routes). We show that the problem is NP-hard in polygons with holes and we present an O( n loglog n) algorithm to find a shortest route in simple rectilinear polygons.

Fast algorithm for polygon decomposition

An O(klog(k)+n) algorithm is developed, where n is the number of versions, to decompose rectilinear polygons into rectangles. This algorithm uses horizontal cuts only and reports nonoverlapping rectangles the union of which is the original rectilinear polygon. This algorithm has been programmed in Pascal on an Apollo DN320 workstation. Experimentation with rectilinear polygons from VLSI artwork indicate that the present algorithm is significantly faster than the plane sweep algorithm and the algorithm proposed by K.D. Gourley and D.M. Green (1983).< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Giant desiccation fissures filled with calcareous eolian sand, Hermosa Formation (Pennsylvanian), southeastern Utah

At two stratigraphic intervals within the upper member of the Upper Pennsylvanian Hermosa Formation, calcareous eolian sand fills downward-tapering fissures that are as much as 18 cm wide and 5.7 m deep. Fissure fillings define orthogonal polygons 10 m or more in diameter. One of the host beds is primarily composed of subtidally deposited limestone, the other is a thinly laminated, nonmarine red siltstone. Both systems of fissure fillings are directly overlain and underlain by large-scale cross-stratified, calcareous eolianites. The limestone host bed contains chert pseudomorphs after gypsum. Compaction of host rocks contorted fissure fillings and caused doming of eolian strata over each fissure. Platy mineral grains in fissure fillings are aligned subparallel to bedding in the host rocks, supporting the view that the fissures were passively filled rather than forcefully injected. These ancient fissure systems are similar in scale and pattern to those that define giant desiccation polygons in numerous Great Basin playas. The Pennsylvanian fissures, like their Holocene counterparts, probably formed when groundwater tables dropped from shallow levels within fine-grained, impermeable deposits into underlying aquifers, greatly decreasing the extent of the capillary fringe. Our study of the fissures and host rocks supports the hypothesis that carbonate grains within the eolianites were deflated from uncemented marine sediments that were broadly exposed during regressive intervals.

A time- and space-optimal algorithm for Boolean mask operations for orthogonal polygons

A time- and space-optimal algorithm for Boolean mask operations for orthogonal polygons

Partitioning and separating sets of orthogonal polygons

We consider the computation of the orthogonal convex hull of a set of orthogonal simple polygons, where by orthogonal we mean that only horizontal and vertical orientations of edges are allowed. This, as we show, induces a partition of the set of orthogonal polygons. We provide an algorithm to compute the orthogonal-convex-hull partition of a set of p orthogonal polygons in O( n log p) time and O( n) space, where n is the total number of vertices of the p polygons. Moreover we prove that this is both time and space optimal. For the case p = 1, an O( n) time- and space-optimal algorithm is also presented. We simplify the description of the algorithms by making essential use of a decomposition theorem which we also prove. As we demonstrate, these results enable us to solve a number of separability problems for polygons. In particular, the group 4-way isoseparability of p polygons with a total of n vertices can be solved in O( n log p) time and O( n) space.

Optimal computation of finitely oriented convex hulls

We define four versions of the “convex hull” of a simple finitely oriented polygon (i.e., a polygon whose edge orientations all belong to some fixed finite set of angles) and give optimal algorithms to find them. Two of these generalize the notions of the orthogonal convex hull of an orthogonal polygon and the traditional “bounding box” of a polygon. Three of the hulls have worst-case time complexity Θ(n + f) and worst case space complexity θ(n) space, where n is the number of edges of a given polygon and f (≥2) is the number of allowed orientations. We also show that testing whether an arbitrary simple polygon is (finitely oriented) convex has worst-case time and space complexity θ(n + f) and θ(n), respectively.

Systolic algorithms for rectilinear polygons

We develop systolic algorithms for the OR, AND, Oversizing, and Undersizing of rectilinear polygons. These algorithms work on an edge representation of the polygons rather than on a bit map representation. The algorithms are to be run on a systolic chain of processors. The edges are input at the left end of this chain. From here, they ‘float’ as far to the right as necessary. As edges float to the right, they compare themselves with edges that are resident in the processors they are floating through. During this comparison the output polygons are generated. Output polygons float to the left. These polygons are output from the left end of the chain. The throughput of the systolic system can be improved by increasing the length of the processor chain.

SLS: An Advanced Symbolic Layout System for Bipolar and FET Design

This paper describes features of the Symbolic Layout System (SLS), an advanced layout system for VLSI designs. Symbolic layout is a method by which point objects, wires, and regions are sketched on a virtual grid, converted to shapes, and then spaced according to a set of minimum ground rules using a compactor. Point objects are defined as terminals, FET transistors, contacts, or vias. These objects are connected using wires or sticks. Flexible regions (orthogonal polygons) are provided for building wells, bipolar transistors, or complex devices. Network extraction is performed on the wires and point objects to check connectivity and provide net information to the compactor. The compaction process is interactively controlled, allowing the user to override design rules, insert jogs, build compaction fences, and specify the compaction center. SLS is technology independent, making it useful for bipolar and FET technologies.

Numerical method based on conformal transformations for calculating resistances in integrated circuits

ABSTRACT In this work we present a procedure for calculating resistances in rectangles with two external contacts which is based on the Schwarz-Christoffel conformal transformation. The elliptic integrals and functions resulting from this transformation are calculated by means of polynomial and series expansion approximations, incorporated in a computer program that has been developed. This program proves to be very accurate, with execution times in the order of a few milliseconds per figure. In orthogonal polygons this method of calculating resistances is an effective complement to the technique based on breaking the original figure into simpler regions.

An algorithm for covering polygons with rectangles

Decomposing a polygon into simple shapes is a basic problem in computational geometry, with applications in pattern recognition and integrated circuit manufacture. Here we examine the special case of covering a rectilinear polygon (or polyomino) with the minimum number of rectangles, with overlapping allowed. The problem is NP -hard. However, we give here an O ( v 2 ) algorithm for constructing a minimum rectangle cover, when the polygon is vertically convex. (Here v is the number of vertices.) The problem is first reduced to a 1-dimensional interval “basis” problem. In showing our algorithm produces an optimal cover we give a new proof of a minimum basis-maximum independent set duality theorem first proved by E. Györi ( J. Combin Theory Ser. B 37 , No. 1, 1–9).

The contour problem for rectilinear polygons

The union of a set of p, not necessarily disjoint, rectilinear polygons in the plane determines a set of disjoint rectilinear polygons. We present an O(n log n + e) time and O(n) space algorithm to compute the edges of the disjoint polygons, that is, the contour, where n is the total number of vertices in the original polygons and e the total number in the resulting set. This time-and space-optimal algorithm uses the scan-line paradigm as in two previous approaches to this problem for rectangles, but requires a simpler data structure. Moreover, if the given rectilinear polygons are rectilinear convex, the space requirement is reduced to O(p).

On the definition and computation of rectilinear convex hulls

Recently the computation of the rectilinear convex hull of a collection of rectilinear polygons has been studied by a number of authors. From these studies three distinct definitions of rectilinear convex hulls have emerged. We examine these three definitions for point sets in general, pointing out some of their consequences, and we give optimal algorithms to compute the corresponding rectilinear convex hulls of a finite set of points in the plane.