Simple Orthogonal Polygons Research Articles

Optimizing generalized kernels of polygons

Let \(\mathcal {O}\) be a set of k orientations in the plane, and let P be a simple polygon in the plane. Given two points p, q inside P, we say that p \(\mathcal {O}\)-sees q if there is an \(\mathcal {O}\)-staircase contained in P that connects p and q. The \(\mathcal {O}\)-Kernel of the polygon P, denoted by \(\mathcal {O}\)-\(\mathrm{Kernel }(P)\), is the subset of points of P which \(\mathcal {O}\)-see all the other points in P. This work initiates the study of the computation and maintenance of \(\mathcal {O}\)-\(\mathrm{Kernel }(P)\) as we rotate the set \(\mathcal {O}\) by an angle \(\theta \), denoted by \(\mathcal {O}\)-\(\mathrm{Kernel }_{\theta }(P)\). In particular, we consider the case when the set \(\mathcal {O}\) is formed by either one or two orthogonal orientations, \(\mathcal {O}=\{0^\circ \}\) or \(\mathcal {O}=\{0^\circ ,90^\circ \}\). For these cases and P being a simple polygon, we design efficient algorithms for computing the \(\mathcal {O}\)-\(\mathrm{Kernel }_{\theta }(P)\) while \(\theta \) varies in \([-\frac{\pi }{2},\frac{\pi }{2})\), obtaining: (i) the intervals of angle \(\theta \) where \(\mathcal {O}\)-\(\mathrm{Kernel }_{\theta }(P)\) is not empty, (ii) a value of angle \(\theta \) where \(\mathcal {O}\)-\(\mathrm{Kernel }_{\theta }(P)\) optimizes area or perimeter. Further, we show how the algorithms can be improved when P is a simple orthogonal polygon. In addition, our results are extended to the case of a set \(\mathcal {O}=\{\alpha _1,\dots ,\alpha _k\}\).

Clearing an orthogonal polygon to find the evaders

In a multi-robot system, a number of autonomous robots would sense, communicate, and decide to move within a given domain to achieve a common goal. In the pursuit-evasion problem, a polygonal region is given and a robot called a pursuer tries to find some mobile targets called evaders. The goal of this problem is to design a motion strategy for the pursuer such that it can detect all the evaders. In this paper, we consider a new variant of the pursuit-evasion problem in which the robots (pursuers) each moves back and forth along an orthogonal line segment inside a simple orthogonal polygon P. We assume that P includes unpredictable, moving evaders that have bounded speed. We propose the first motion-planning algorithm for a group of robots, assuming that they move along the pre-located line segments with a constant speed to detect all the evaders with bounded speed. Also, we prove an upper bound for the length of the paths that all pursuers move in the proposed algorithm.

Tight bounds for conflict-free chromatic guarding of orthogonal art galleries

The chromatic art gallery problem asks for the minimum number of “colors” t so that a collection of point guards, each assigned one of the t colors, can see the entire polygon subject to some conditions on the colors visible to each point. In this paper, we explore this problem for orthogonal polygons using orthogonal visibility—two points p and q are mutually visible if the smallest axis-aligned rectangle containing them lies within the polygon. Our main result establishes that for a conflict-free guarding of an orthogonal n-gon, in which at least one of the colors seen by every point is unique, the number of colors is in the worst case Θ(log⁡log⁡n). By contrast, the best known upper bound for orthogonal polygons under standard (non-orthogonal) visibility is O(log⁡n) colors. We also show that the number of colors needed for strong guarding of simple orthogonal polygons, where all the colors visible to a point are unique, is, again in the worst case, Θ(log⁡n). Finally, our techniques also help us establish the first non-trivial lower bound of Ω(log⁡log⁡n/log⁡log⁡log⁡n) for conflict-free guarding under standard visibility. To this end we introduce and utilize a novel discrete combinatorial structure called multicolor tableau.

Guarding monotone art galleries with sliding cameras in linear time

A sliding camera in an orthogonal polygon P—that is, a polygon all of whose edges are axis-parallel—is a point guard g that travels back and forth along an axis-parallel line segment s inside P. A point p in P is guarded by g if and only if there exists a point q on s such that line segment pq is normal to s and contained in P. In the minimum sliding cameras (MSC) problem, the objective is to guard P with the minimum number of sliding cameras.We give a dynamic programming algorithm that solves the MSC problem exactly on monotone orthogonal polygons in O(n) time, improving the 2-approximation algorithm of Katz and Morgenstern (2011). More generally, our algorithm can be used to solve the MSC problem in O(n) time on simple orthogonal polygons P for which the dual graph induced by the vertical decomposition of P is a path. Our results provide the first polynomial-time exact algorithms for the MSC problem on a non-trivial subclass of orthogonal polygons.

화랑 문제의 최소 이동 경비원 수 알고리즘

Abstract Given art gallery  with  vertices, the maximum (sufficient) number of mobile guards is ⌊⌋ for simple polygon and ⌊⌋ for simple orthogonal polygon. However, there is no polynomial time algorithm for minimum number of mobile guards. This paper suggests polynomial time algorithm for the minimum number of mobile guards. Firstly, we obtain the visibility graph which is connected all edges if two vertices can be visible each other. Secondly, we select vertex  with  and  with  in    and delete visible edges from  and incident edges. Thirdly, we select   in partial graphs and select edges that is the position of mobile guards. This algorithm applies various art galley problems with simple polygons and orthogonal polygons art gallery. As a results, the running time of proposed algorithm is linear time complexity and can be obtain the minimum number of mobile guards. Key Words : Art Gallery Problem, Polygon, Orthogonal Polygon, Stationary Guard, Mobile Guard

Exact solutions and bounds for general art gallery problems

The classical Art Gallery Problem asks for the minimum number of guards that achieve visibility coverage of a given polygon. This problem is known to be NP-hard, even for very restricted and discrete special cases. For the case of vertex guards and simple orthogonal polygons, Cuoto et al. have recently developed an exact method that is based on a set-cover approach. For the general problem (in which both the set of possible guard positions and the point set to be guarded are uncountable), neither constant-factor approximation algorithms nor exact solution methods are known. We present a primal-dual algorithm based on linear programming that provides lower bounds on the necessary number of guards in every step and—in case of convergence and integrality—ends with an optimal solution. We describe our implementation and give experimental results for an assortment of polygons, including nonorthogonal polygons with holes.

On finding an orthogonal convex skull of a digital object

AbstractA combinatorial algorithm to compute an orthogonal convex skull of a digital object imposed on the background grid is presented in this paper. The proposed algorithm has the time complexity of O(n log n), which improves the earlier method of O(n2) time complexity for finding the convex skull of a simple orthogonal polygon. A set of rules is formulated first and then an orthogonal convex skull is derived by applying these rules while traversing along the boundary of the inner orthogonal polygon that tightly inscribes the given digital object. The algorithm uses only comparison and addition in the integer domain, which makes it amenable to fast real‐world applications. Experimental results on different shapes have also been presented to demonstrate the efficacy and elegance of the proposed technique. © 2011 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 21, 14–27, 2011.

On the tileability of polygons with colored dominoes

Analysis of Algorithms We consider questions concerning the tileability of orthogonal polygons with colored dominoes. A colored domino is a rotatable 2 × 1 rectangle that is partitioned into two unit squares, which are called faces, each of which is assigned a color. In a colored domino tiling of an orthogonal polygon P, a set of dominoes completely covers P such that no dominoes overlap and so that adjacent faces have the same color. We demonstrated that for simple layout polygons that can be tiled with colored dominoes, two colors are always sufficient. We also show that for tileable non-simple layout polygons, four colors are always sufficient and sometimes necessary. We describe an O(n) time algorithm for computing a colored domino tiling of a simple orthogonal polygon, if such a tiling exists, where n is the number of dominoes used in the tiling. We also show that deciding whether or not a non-simple orthogonal polygon can be tiled with colored dominoes is NP-complete.

A LINEAR-TIME ALGORITHM FOR COVERING SIMPLE POLYGONS WITH SIMILAR RECTANGLES

We study the problem of covering a simple orthogonal polygon with a minimum number of (possibly overlapping) squares, all internal to the polygon. The problem has applications in VLSI mask generation, incremental update of raster displays, and image compression. We give a linear time algorithm for covering a simple polygon, specified by its vertices, with squares. Covering with similar rectangles (having a given x/y ratio) is an equivalent problem.

Placing guards inside orthogonal polygons

We consider the problem of locating guards inside a polygon under standard visibility and orthogonal visibility. Given a polygon, the maximum visibility problem ( MaxVP) asks for locating a point inside the polygon from which the visible area is maximized. Minimum visibility problem ( MinVP) is defined similarly. Under standard visibility, we show that the boundary point solution for MinVP can be obtained in O( n 3) time for simple orthogonal polygons and in O( n 3log n) time for orthogonal polygons with holes. For class 3 polygons under stair-case visibility, we show that the solution for MinVP lies on the boundary; this leads to an O( n 2) time algorithm to locate the solution.

Recognizing s-star polygons

We consider the problem of recognizing star-polygons under staircase visibility ( s-visibility). We show that the s-visibilitypolygon from a point inside a simple orthogonal polygon of n sides can be computed in O( n) time. When the polygon contains holes the algorithm runs in O( n log n) time, which we prove to be optimal by linear time reduction from sorting. We present an algorithm for computing the s-kernel of a polygon in O( n) time for simple orthogonal polygons and in O( n 2) time for orthogonal polygons with holes; both complexities are optimal in the worst case. Finally, we report the main result of this paper: we show that even though the s-kernel of a polygon with holes may have Ω(n 2) components it is possible to recognize such polygons in O( n log n) time.

Covering grids and orthogonal polygons with periscope guards

The problem of finding minimum guard covers is NP-hard for simple polygons and open for simple orthogonal polygons. Alternative definitions of visibility have been considered for orthogonal polygons. In this paper we try to determine the complexity of finding guard covers in orthogonal polygons by considering periscope visibility. Under periscope visibility, two points in an orthogonal polygon are visible if there is an orthogonal path with at most one bend that connects them without intersecting the exterior of the polygon. We show that finding minimum periscope guard (as well as k-periscope and s-guard) covers is NP-hard for 3-d grids. We present an O( n 3) algorithm for finding minimum periscope guard covers for simple grids and discuss how to extend the algorithm to obtain minimum k-periscope guard covers. We show that this algorithm can be applied to obtain minimum periscope guard covers for a class of simple orthogonal polygon in O( n 3).

On graphs preserving rectilinear shortest paths in the presence of obstacles

In physical VLSI design, network design (wiring) is the most time-consuming phase. For solving global wiring problems, we propose to first compute from the layout geometry a graph that preserves all shortest paths between pairs of relevant points, and then to operate on that graph for computing shortest paths, Steiner minimal tree approximations, or the like. For a set of points and a set of simple orthogonal polygons as obstacles in the plane, withn input points (polygon corner or other) altogether, we show how a shortest paths preserving graph of sizeO(n logn) can be computed in timeO(n logn) in the worst case, with spaceO(n). We illustrate the merits of this approach with a simple example: If the length of a longest edge in the graph is bounded by a polynomial inn, an assumption that is clearly fulfilled for graphs derived from VLSI layout geometries, then a shortest path can be computed in timeO(n logn log logn) in the worst case; this result improves on the known best one ofO(n(logn)3/2).

An efficient algorithm for finding a two-pair, and its applications

Let G = (V, E) be an undirected, finite, and simple graph. All the definitions not given here may be found in [l]. A pair (u, o) of nonadjacent vertices in G form a two-pair if every chordless path between u and v is of length two. We shall digress briefly to remark on the applications of our problem. One of the fundamental notions in computational geometry is the concept of visibility. A visibility graph has vertices which correspond to geometric components such as points or lines, and edges which correspond to visibility of these components from each other. O’Rourke [5] declares that the fundamental problems involving visibility in computational geometry will not be solved until the combinatorial structure of the visibility graphs is more fully understood. There is an intimate connection between the visibility problem and the polygon covering problem. Recently, several interesting structural characterizations of visibility graphs for regions inside a simple orthogonal polygon were reported in [3, 41. In particular, the visibility graph of a special type of orthogonal polygon called Class 3 polygon [3] is a weakly triangulated graph. It is also shown in [3, 41 that the polygon covering problem for the Class 3 polygon reduces to the clique covering problem on weakly triangulated graphs. The fact that weakly triangulated graphs are perfect is used to derive several insightful duality relationships for Class 3 polygon.

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.

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.

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.