Orthogonal Convex Research Articles

Computing minimum-area rectilinear convex hull and L-shape

We study the problems of computing two non-convex enclosing shapes with the minimum area; the L-shape and the rectilinear convex hull. Given a set of n points in the plane, we find an L-shape enclosing the points or a rectilinear convex hull of the point set with minimum area over all orientations. We show that the minimum enclosing shapes for fixed orientations change combinatorially at most O ( n ) times while rotating the coordinate system. Based on this, we propose efficient algorithms that compute both shapes with the minimum area over all orientations. The algorithms provide an efficient way of maintaining the set of extremal points, or the staircase, while rotating the coordinate system, and compute both minimum enclosing shapes in O ( n 2 ) time and O ( n ) space. We also show that the time complexity of maintaining the staircase can be improved if we use more space.

On Constructing the Minimum Orthogonal Convex Polygon for the Fault-Tolerant Routing in 2-D Faulty Meshes

The rectangular faulty block model is the most commonly used fault model for designing fault-tolerant, and deadlock-free routing algorithms in mesh-connected multicomputers. The convexity of a rectangle facilitates simple, efficient ways to route messages around fault regions using relatively few or no virtual channels to avoid deadlock. However, such a faulty block may include many nonfaulty nodes which are disabled, i.e., they are not involved in the routing process. Therefore, it is important to define a fault region that is convex, and at the same time, to include a minimum number of nonfaulty nodes. In this paper, we propose an optimal solution that can quickly construct a set of minimum faulty polygons, called orthogonal convex polygons, from a given set of faulty blocks in a 2-D mesh (or 2-D torus). The formation of orthogonal convex polygons is implemented using either a centralized, or distributed solution. Both solutions are based on the formation of faulty components, each of which consists of adjacent faulty nodes only, followed by the addition of a minimum number of nonfaulty nodes to make each component a convex polygon. Extensive simulation has been done to determine the number of nonfaulty nodes included in the polygon, and the result obtained is compared with the best existing known result. Results show that the proposed approach can not only find a set of minimum faulty polygons, but also does so quickly in terms of the number of rounds in the distributed solution.

A Distributed Formation of Smallest Faulty Orthogonal Convex Polygons in 2-D Meshes

The rectangular faulty block model is the most commonly used fault model for designing fault-tolerant and deadlock-free routing algorithms in mesh-connected multicomputers. The convexity of a rectangle facilitates simple and efficient ways to route messages around fault regions using relatively few virtual channels to avoid deadlock. However, such a faulty block may include many nonfaulty nodes which are disabled, i.e., they are not involved in the routing process. Therefore, it is important to define a fault region that is convex, and at the same time, to include a minimum number of nonfaulty nodes. In this paper, we propose a simple and efficient distributed algorithm that can quickly construct a set of special convex polygons, called orthogonal convex polygons, from a given set of rectangular faulty blocks in a 2-D mesh (or 2-D torus). The formation of orthogonal convex polygons is done through a labeling scheme based on iterative message exchanges among neighboring nodes. For a given faulty block, after some nonfaulty nodes have been removed, the block is split into a set of orthogonal convex polygons, each of which is the smallest orthogonal convex polygon that contains all the faults it covers. Moreover, we show that the number of nonfaulty nodes covered in these polygons is no more than the one in the smallest orthogonal convex polygon that includes all the faulty nodes in the original faulty block. Finally, we discuss one open problem and present extensions to multidimensional meshes.

VISIBILITY GRAPHS OF STAIRCASE POLYGONS WITH UNIFORM STEP LENGTH

Let Γn denote the collection of visibility graphs of staircase polygons (orthogonal convex fans) which consist of n−1 horizontal steps of arbitrary lengths. We show that for n≥7 there are graphs in Γn which cannot be realized as the visibility graphs of staircase polygons with uniform step length.

A linear-time algorithm to construct a rectilinear Steiner minimal tree fork-extremal point sets

Ak-extremal point set is a point set on the boundary of ak-sided rectilinear convex hull. Given ak-extremal point set of sizen, we present an algorithm that computes a rectilinear Steiner minimal tree in timeO(k 4 n). For constantk, this algorithm runs inO(n) time and is asymptotically optimal and, for arbitraryk, the algorithm is the fastest known for this problem.

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.

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.

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.