Generating Empty Convex Polygon Randomly from a Subset of Given Point Set

The convex hull of a planer set of points can be defined as the set of vertices of the smallest convex polygon containing all the points. If S is a planer set of points then convex layers of S can be derived by iteratively computing the convex hull of S and then removing it from S. Computation of the convex layers has widely been studied in the static environment where the set of points is fixed. The theoretical lower bound of computing the convex layers of a fixed set of points is O (n log n). The static convex layers algorithms with the optimal time complexity have already been proposed in the literature. A set of points where the points can be inserted or deleted is called a dynamic set of points. The set of convex layers should be reconstructed from the scratch at each insertion or deletion if a static convex layers algorithm was used to handle a dynamic set of points. Therefore, it takes O (n log n) time to handle an insertion or a deletion even for an optimal static convex layers algorithm. A dynamic convex layers algorithm has been proposed recently that can perform an insertion or a deletion by doing a slight modification to the existing set of convex layers. It takes O (n) time for an insertion or a deletion. It assumes that the set of points does not contain collinear points and that assumption is not valid in practical applications. Furthermore, the notion of tangent used in that approach restricts the extension of the algorithm into higher dimensions. This paper proposes a novel dynamic convex layers algorithm to eliminate the drawbacks in the existing algorithm. Salient feature of this algorithm is that it represents each layer as a set of line segments. The layers are modified upon an insertion or a deletion of a point by adding some new line segments and deleting some existing line segments. The proposed algorithm takes O (n 3 /k 2 ) time for an insertion or a deletion where k is the number of convex layers. A computer implementation is also presented. The proposed algorithm can work with set of points with collinear points and coincident points. The notion used in the algorithm can easily be extended to higher dimensions. Suppose the set of convex layers have already been found for a given set of points. Further, suppose that the layers are close to each other and new points are expected to fall within the region bounded by the outermost layer. This kind of situations widely occurs in practice and then the proposed algorithm takes O (n) time for an inclusion or a deletion.

Traditional geometric algorithms are often presented as if input imprecision does not exist, even though it is often unavoidable. We argue that in some cases, it may be desirable for geometric algorithms to treat this imprecision as an explicit component of the input, and to reflect this imprecision in the output. Starting with three problems from computational geometry whose inputs are planar point sets (Voronoi diagrams, convex hulls, and smallest bounding discs), we recast these as problems where each input point’s location is imprecise, but known to lie within a particular region of uncertainty. Where algorithms to solve each of the original problems produce a single geometric object as output, the algorithms that we present typically produce either guaranteed or possible output objects. A guaranteed object represents qualities that can be guaranteed for every point set that is consistent with the uncertain regions, and a possible object represents qualities that exist for at least one such point set. By dealing with input imprecision explicitly, these guaranteed and possible objects can represent a more accurate view of what can be reliably inferred from the input data.

Expected size of random Tukey layers and convex layers

In the paper, we present and prove Castle Theorem of Convex Hull, design and realize the Fast Rampart Searching Algorithm. The algorithm can be treated as the preprocess of Convex Hull calculation of very large planar point set. When calculating the Convex Hull of a very large planar point set, we can use the Fast Rampart Searching Algorithm to get a very small part points as candidate point set, and then we can get the Convex Hull of the whole planar point set from the candidate point set through other algorithms. Keywords-fast rampart searching algorithm; castle theorem; convex hull; computational geometry

Random polygon Generation from a given set of points is very important for different applications. A heuristic termed as “GRP_AS” has been proposed here to generate a random simple polygon from a given set of `n' points in 2-Dimensional plane. The “2-Opt Move” heuristic with time complexity O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> ) is the best known among the existing heuristics to generate a simple polygon. We recently designed another heuristic “GRP_CH” for the same problem with time complexity O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ) which was shown to be better than that of “2-Opt Move” heuristic in terms of random behaviour and time complexity. In this work, our main objective is to design an algorithm with reduced time complexity. The proposed heuristic, “GRP_AS” is based on the principle of random selection of a line segment and angular sorting of the point set. This “GRP_AS” heuristic takes O(nlogn) time which is less than that of “2-Opt Move” heuristic as well as “GRP_CH” heuristic. The lower bound and upper bound on the number of polygons generated in “GRP_AS” heuristic has been computed.

Matching geometric objects with respect to their Hausdorff distance is a well investigated problem in Computational Geometry with various application areas. The variant investigated in this paper is motivated by the problem of determining a matching (in this context also called registration) for neurosurgical operations. The task is, given a sequence \(\mathcal{P}\) of weighted point sets (anatomic landmarks measured from a patient), a second sequence \(\mathcal{Q}\) of corresponding point sets (defined in a 3D model of the patient) and a transformation class \(\mathcal{T}\), compute the transformations \(t\in\mathcal{T}\) that minimize the weighted directed Hausdorff distance of \(t(\mathcal{P})\) to \(\mathcal{Q}\). The weighted Hausdorff distance, as introduced in this paper, takes the weights of the point sets into account. For this application, a weight reflects the precision with which a landmark can be measured.We present an exact solution for translations in the plane, a simple 2-approximation as well as a FPTAS for translations in arbitrary dimension and a constant factor approximation for rigid motions in the plane or in ℝ3.

Structuring finite sets of points is at the heart of computational geometry. Such point sets arise naturally in many applications. Examples in R3 are point sets sampled from the surface of a solid or the locations of atoms in a molecule. A first step in processing these point sets is to organize them in some data structure. Structuring a point set into a simplicial complex like the Delaunay triangulation has turned out to be appropriate for many modeling tasks. Here we introduce the flow complex which is another simplicial complex that can be computed efficiently from a finite set of points. The flow complex turned out to be well suited for surface reconstruction from a finite sample and for some tasks in structural biology. Here we study mathematical and algorithmic properties of the flow complex and show how to exploit it in applications.

Polyhedral and Algebraic Methods in Computational Geometry provides a thorough introduction into algorithmic geometry and its applications. It presents its primary topics from the viewpoints of discrete, convex and elementary algebraic geometry. The first part of the book studies classical problems and techniques that refer to polyhedral structures. The authors include a study on algorithms for computing convex hulls as well as the construction of Voronoi diagrams and Delone triangulations. The second part of the book develops the primary concepts of (non-linear) computational algebraic geometry. Here, the book looks at Grbner bases and solving systems of polynomial equations. The theory is illustrated by applications in computer graphics, curve reconstruction and robotics. Throughout the book, interconnections between computational geometry and other disciplines (such as algebraic geometry, optimization and numerical mathematics) are established. Polyhedral and Algebraic Methods in Computational Geometry is directed towards advanced undergraduates in mathematics and computer science, as well as towards engineering students who are interested in the applications of computational geometry.

Spatial data structures form a core ingredient of many geometric algorithms, both in theory and in practice. Many of these data structures, especially the ones used in practice, are based on partitioning the underlying space (examples are binary space partitions and decompositions of polygons) or partitioning the set of objects (examples are bounding-volume hierarchies). The efficiency of such data structures---and, hence, of the algorithms that use them---depends on certain characteristics of the partitioning. For example the performance of many algorithms that use binary space partitions (BSPs) depends on the size of the BSPs. Similarly, the performance of answering range queries using bounding-volume hierarchies (BVHs) depends on the so-called crossing number that can be associated with the partitioning on which the BVH is based. Much research has been done on the problem of computing partitioning whose characteristics are good in the worst case. In this thesis, we studied the problem from a different point of view, namely instance-optimality. In particular, we considered the following question: given a class of geometric partitioning structures---like BSPs, simplicial partitions, polygon triangulations, …---and a cost function---like size or crossing number---can we design an algorithm that computes a structure whose cost is optimal or close to optimal for any input instance (rather than only worst-case optimal). We studied the problem of finding optimal data structures for some of the most important spatial data structures. As an example having a set of n points and an input parameter r, It has been proved that there are input sets for which any simplicial partitions has crossing number ¿(vr). It has also been shown that for any set of n input points and the parameter r one can make a simplicial partition with stabbing number O(vr). However, there are input point sets for which one can make simplicial partition with lower stabbing number. As an example when the points are on a diagonal, one can always make a simplicial partition with stabbing number 1. We started our research by studying BSPs for line segments in the plane, where the cost function is the size of the BSPs. A popular type of BSPs for line segments are the so-called auto-partitions. We proved that finding an optimal auto-partition is NP-hard. In fact, finding out if a set of input segments admits an auto-partition without any cuts is already NP-hard. We also studied the relation between two other types of BSPs, called free and restricted BSPs, and showed that the number of cuts of an optimal restricted BSP for a set of segments in R2 is at most twice the number of cuts of an optimal free BSP for that set. The details are being represented in Chapter 1 of the thesis. Then we turned our attention to so-called rectilinear r-partitions for planar point sets, with the crossing number as cost function. A rectilinear r-partition of a point set P is a partitioning of P into r subsets, each having roughly |P|/r points. The crossing number of the partition is defined using the bounding boxes of the subsets; in particular, it is the maximum number of bounding boxes that can be intersected by any horizontal or vertical line. We performed some theoretical as well as experimental studies on rectilinear r-partitions. On the theoretical side, we proved that computing a rectilinear r-partition with optimal stabbing number for a given set of points and parameter r is NP-hard. We also proposed an exact algorithm for finding optimal rectilinear r-partitions whose running time is polynomial when r is a constant, and a faster 2-approximation algorithm. Our last theoretical result showed that considering only partitions whose bounding boxes are disjoint is not sufficient for finding optimal rectilinear r-partitions. On the experimental side, we performed a comparison between four different heuristics for constructing rectilinear r-partitions. The so-called windmill KD-tree gave the best results. Chapter 2 of the thesis describes all the details of our research on rectilinear r-partitions. We studied another spatial data structure in Chapter 3 of the thesis. Decomposition of the interior of polygons is one of the fundamental problems in computational geometry. In case of a simple polygon one usually wants to make a Steiner triangulation of it, and when we have a rectilinear polygon at hand, one typically wants to make a rectilinear decomposition for it. Due to this reason there are algorithms which make Steiner triangulations and rectangular decompositions with low stabbing number. These algorithms are worst-case optimal. However, similar to the two previous data structures, there are polygons for which one can make decompositions with lower stabbing numbers. In 3 we proposed a 3-approximation for finding an optimal rectangular decomposition of a rectilinear polygon. We also proposed an O(1)-approximation for finding optimal Steiner triangulation of a simple polygon. Finally, in Chapter 4 of the thesis, we considered another optimization problem, namely how to approximate a piecewise-linear function F: R ¿R with another piecewise-linear function with fewer pieces. Here one can distinguish two versions of the problem. The first one is called the min-k problem; the goal is then to approximate the function within a given error e such that the resulting function has the minimum number of links. The second one is called the min-e problem; here the goal is to find an approximation with at most k links (for a given k) such that the error is minimized. These problems have already been studied before. Our contribution is to consider the problem for so-called uncertain functions, where the value of the input function F at its vertices is given as a discrete set of different values, each with an associated probability. We show how to compute an approximation that minimizes the expected error.

We consider the problem of finding the staircase kernel in orthogonal polygons, with or without holes, in the plane. Orthogonal polygon is a simple polygon in the plane whose sides are either horizontal or vertical. We generalize the notion of visibility in the following way: We say that two points a and b in an orthogonal polygon P are visible to each other via staircase paths if and only if there exist an orthogonal chain connecting a and b and lying entirely in the interior of P. Furthermore, the orthogonal chain should have the property that the angles between the consecutive segments in the chain are either +90 ∘ or −90 ∘ , and these should alternate along the chain. There are two principal types of staircases, NW-SE and NE-SW. The notion of staircase visibility has been studied in the literature for the last three decades. Based on this notion we can generalize the notion of star-shapedness. A polygon P is called star-shaped under staircase visibility, or simply s-star if and only if there is nonempty set of points S in the interior of P, such that any point of S sees any point of P via staircase path. The largest such set of points is called the staircase kernel of P and denoted ker P. Our work is motivated by the work of Breen (1). She proves that the staircase kernel of an orthogonal polygon without holes is the intersection of all maximal orthogonally convex polygons contained in it. We extend Breen's results for the case when the orthogonal polygon has holes. We prove the necessary geometric properties, and use them to derive a quadratic time, O(���� 2 ) algorithm for computing the staircase kernel of an orthogonal polygon with holes, having n vertices in total, including the holes' vertices. The algorithm is based on the plane sweep technique, widely used in Computational Geometry(4). Our result is optimal in the case of orthogonal polygon with holes, since the kernel (as proven) can consist of quadratic number of disjoint regions. In the case of polygon without holes, there is a linear time algorithm by Gewali(3), that is specific to the case of a polygon without holes. We present ex- amples of our algorithm's results.

We give the first differentially private algorithms that estimate a variety of geometric features of points in the Euclidean space, such as diameter, width, volume of convex hull, min-bounding box, min-enclosing ball, etc. Our work relies heavily on the notion of Tukey-depth. Instead of (non-privately) approximating the convex-hull of the given set of points P, our algorithms approximate the geometric features of D_{P}(κ) - the κ-Tukey region induced by P (all points of Tukey-depth κ or greater). Moreover, our approximations are all bi-criteria: for any geometric feature μ our (α,Δ)-approximation is a value sandwiched between (1-α)μ(D_P(κ)) and (1+α)μ(D_P(κ-Δ)). Our work is aimed at producing a (α,Δ)-kernel of D_P(κ), namely a set 𝒮 such that (after a shift) it holds that (1-α)D_P(κ) ⊂ CH(𝒮) ⊂ (1+α)D_P(κ-Δ). We show that an analogous notion of a bi-critera approximation of a directional kernel, as originally proposed by [Pankaj K. Agarwal et al., 2004], fails to give a kernel, and so we result to subtler notions of approximations of projections that do yield a kernel. First, we give differentially private algorithms that find (α,Δ)-kernels for a fat Tukey-region. Then, based on a private approximation of the min-bounding box, we find a transformation that does turn D_P(κ) into a fat region but only if its volume is proportional to the volume of D_P(κ-Δ). Lastly, we give a novel private algorithm that finds a depth parameter κ for which the volume of D_P(κ) is comparable to the volume of D_P(κ-Δ). We hope our work leads to the further study of the intersection of differential privacy and computational geometry.

Let P be a set of points in the plane and assume that P is in general position. A triangulation of P is a maximal non-crossing geometric graph with vertex set P. All bounded faces of a triangulation are triangles (that's why we call them triangulation), the unbounded face is the outside of the convex hull of P. Triangulations play a prominent role in many applicable and applied disciplines like computational geometry, computer graphics and numerical modeling. In this chapter we discuss combinatorial and geometrical properties of triangulations and of the flip-graph on the set of all triangulations of a point set P.

Solvers for semidefinite programming (SDP) have evolved a great deal in the last decade, and their development continues. In order to further support and encourage this development, we present a new test set of SDP instances. These instances arise from recent applications of SDP in the coding theory, computational geometry, graph theory and structural design. Most of these instances have a special structure that may be exploited during a pre-processing phase, e.g. algebraic symmetry, or low rank in the constraint matrices.

A polyhedral object in 3-dimensional space is often well represented by a set of points and line segments that act asits features. By a nice viewpoint of an object we mean a projective view in which all (or most) of the features of theobject, relevant for some task, are clearly visible. Such a view is often called a non-degenerate view or projection. Inthis paper we are concerned with computing non-degenerate orthogonal and perspective projections of sets of pointsand line segments (objects) in 3-dimensional space. We outline the areas in which such problems arize, discuss recentresearch on the computational complexity of these problems, illustrate the fundamental ideas used in the design ofalgorithms for computing non-degenerate projections, and provide pointers to the literature where the results can befound.Keywords: Data visualization, nice viewpoints, knot theory, computer vision, graph drawing, computer graphics,degeneracies, general position, orthogonal projections, perspective projections, robustness of algorithms, computa-tional geometry.

Best-first and depth-first heuristic search algorithms often assume underlying search graphs with only nonnegative edge costs and attempt to optimize simple objective functions. Applicability of these algorithms to graphs with both positive and negative edge costs is not completely studied. In the paper, two new problems are identified: one in computational geometry and the other in the layout design of very large scale integrated (VLSI) circuits. The former problem relates to a weight-balanced bipartitioning of a given set of points in a plane. The goal of the second problem is to find an area-balanced staircase path in a VLSI floorplan. Formulations of these problems lead to an interesting directed acyclic search graph with positive, zero and negative edge costs and an objective function of general nature. These problems are NP-hard. To solve such general problems optimally, search schemes are proposed. Experimental results reveal the efficacy and versatility of the proposed schemes, the depth-first scheme being the better choice. It is shown that the classical number-partitioning problem can also be formulated in this framework.