Guarantees Concerning Geometric Objects with Imprecise Points

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

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 study SET COVER for orthants: Given a set of points in a d-dimensional Euclidean space and a set of orthants of the form (-infty,p_1] x ... x (-infty,p_d], select a minimum number of orthants so that every point is contained in at least one selected orthant. This problem draws its motivation from applications in multi-objective optimization problems. While for d=2 the problem can be solved in polynomial time, for d>2 no algorithm is known that avoids the enumeration of all size-k subsets of the input to test whether there is a set cover of size k. Our contribution is a precise understanding of the complexity of this problem in any dimension d >= 3, when k is considered a parameter: - For d=3, we give an algorithm with runtime n^O(sqrt{k}), thus avoiding exhaustive enumeration. - For d=3, we prove a tight lower bound of n^Omega(sqrt{k}) (assuming ETH). - For d >=slant 4, we prove a tight lower bound of n^Omega(k) (assuming ETH). Here n is the size of the set of points plus the size of the set of orthants. The first statement comes as a corollary of a more general result: an algorithm for SET COVER for half-spaces in dimension 3. In particular, we show that given a set of points U in R^3, a set of half-spaces D in R^3, and an integer k, one can decide whether U can be covered by the union of at most k half-spaces from D in time |D|^O(sqrt{k})* |U|^O(1). We also study approximation for SET COVER for orthants. While in dimension 3 a PTAS can be inferred from existing results, we show that in dimension 4 and larger, there is no 1.05-approximation algorithm with runtime f(k)* n^o(k) for any computable f, where k is the optimum.

Let P be a set of n points in ℝ^d in general position. A median hyperplane (roughly) splits the point set P in half. The yolk of P is the ball of smallest radius intersecting all median hyperplanes of P. The egg of P is the ball of smallest radius intersecting all hyperplanes which contain exactly d points of P. We present exact algorithms for computing the yolk and the egg of a point set, both running in expected time O(n^(d-1) log n). The running time of the new algorithm is a polynomial time improvement over existing algorithms. We also present algorithms for several related problems, such as computing the Tukey and center balls of a point set, among others.

This paper establishes a relation between the field of study of Geometric Design of the Structures known as and Computational Geometry. According to some properties of Voronoi Diagrams and Convex Hulls, and by means of two transformations: one inversive, the other projective, a way for generating polyhedra approximating spheres and other second order surfaces (ellipsoids, paraboloids and hyperboloids), starting from a simple Voronoi Diagram, is shown. The paper concludes with the geometric design of C-TANGENT domes: a new way for generating Spatial Meshes that can become a possible field of application for Computational Geometry.

Chapter 7 A survey of computational geometry

The concept of the minimum spanning tree (MST) plays an important role in topological network design, because it models a cheapest connected network. In a tree, however, the failure of a vertex can disconnect the network. In order to tolerate such a failure, we generalize the MST to the concept of a cheapest biconnected network. For a set of points in the Euclidean plane, we show that it is NP-hard to find a cheapest biconnected spanning graph, where edge costs are the Euclidean distances of the respective points. We propose a different type of subgraph, based on forbidding (due to failure) the use of a vertex. A minimum spanning multi-tree is a spanning graph that contains for each possible forbidden vertex a spanning tree that is minimum among the spanning trees that do not use the forbidden vertex. We propose a worst-case time optimal algorithm for computing a minimum spanning multi-tree for a planar Euclidean point set. A minimum spanning multi-tree is cheap, even though it embeds a linear number of MSTs: Its cost is more than the MST cost only by a constant factor. Furthermore, we propose a linear time algorithm for computing a cheap vertex failure tolerant graph, given the Delaunay triangulation. This graph bounds the cost of the minimum spanning multi-tree from above.

Given a set of geometric objects each associated with a time value, we wish to determine whether a given property is true for a subset of those objects whose time values fall within a query time window. We call such problems time-windowed decision problems, and they have been the subject of much recent attention, for instance studied by Bokal, Cabello, and Eppstein [SoCG 2015]. In this paper, we present new approaches to this class of problems that are conceptually simpler than Bokal et al.'s, and also lead to faster algorithms. For instance, we present algorithms for preprocessing for the time-windowed 2D diameter decision problem in O(n log n) time and the time-windowed 2D convex hull area decision problem in O(n alpha(n) log n) time (where alpha is the inverse Ackermann function), improving Bokal et al.'s O(n log^2 n) and O(n log n loglog n) solutions respectively. Our first approach is to reduce time-windowed decision problems to a generalized range successor problem, which we solve using a novel way to search range trees. Our other approach is to use dynamic data structures directly, taking advantage of a new observation that the total number of combinatorial changes to a planar convex hull is near linear for any FIFO update sequence, in which deletions occur in the same order as insertions. We also apply these approaches to obtain the first O(n polylog n) algorithms for the time-windowed 3D diameter decision and 2D orthogonal segment intersection detection problems.

Given a complex polynomial p(z) with at least three distinct roots, we first prove that no rational iteration function exists where the basin of attraction of a root coincides with its Voronoi cell. In spite of this negative result, we prove that the Voronoi diagram of the roots can be well approximated through a high order sequence of iteration functions, the Basic Family, B m (z), m≥2. Let θ be a simple root of p(z), V(θ) its Voronoi cell, and A m (θ) its basin of attraction with respect to B m (z). We prove that given any closed subset C of V(θ), including any homothetic copy of V(θ), there exists m 0 such that for all m≥m 0, C is also a subset of A m (θ). This implies that when all roots of p(z) are simple, the basins of attraction of B m (z) uniformly approximate the Voronoi diagram of the roots to within any prescribed tolerance. Equivalently, the Julia set of B m (z), and hence the chaotic behavior of its iterations, will uniformly lie to within prescribed strip neighborhood of the boundary of the Voronoi diagram. In a sense, this is the strongest property a rational iteration function can exhibit for polynomials. Next, we use the results to define and prove an infinite layering within each Voronoi cell of a given set of points, whether known implicitly as roots of a polynomial equation, or explicitly via their coordinates. We discuss potential application of our layering in computational geometry.

With the flood of information available today the question how to deal with high dimensional data/signals, which are cumbersome to handle, to calculate with and to store, is highly important. One approach to reducing this flood is to find sparse signal representations, as a signal that is the linear combination of a few elements from a pool of building blocks, can be reduced to the few coefficients of this representation. If these building blocks form a basis, finding the sparse representation poses no problem but unfortunately not many signal classes are sparse in a basis. Taking more building blocks, i.e. a redundant dictionary, increases the chances of having sparse representations, but actually finding them becomes very hard. This led to the development of numerous strategies and algorithms for finding sparse representations, with varying complexity and success rate. The first part of the thesis deals with two of those algorithms, Thresholding and Matching Pursuit, from a more theoretical point of view. It is shown that both those greedy algorithms can be improved with a little trick, that does not increase their complexity, and that when considering their average instead of their worst case performance they perform quite well in comparison to more complex methods. The second part of thesis treats questions concerning the whole dictionary and its properties. First it gives more evidence that sparsity is useful by extending the concept of compressed sensing to signals that are sparse not in a basis but in a redundant dictionary. Thus to record a sparse signal it is not necessary to make as many measurements as the dimension of the signal but only a multiple of the number of dictionary elements used to represent it. Next we show that dictionaries cannot only provide sparse representations but that their geometric properties can also be exploited to model data structures. Here we explain how to model different subclasses of a class of signals by incoherent subspaces, present an algorithm to learn a dictionary made out of these subspaces and then use it for classification of faces. Finally we turn back to the sparse representation problem and study the fundamental question how to find a dictionary providing sparse representations. We pick up the idea to learn a dictionary via minimisation of a continuous cost function and provide conditions, guaranteeing that the decomposition of a collection of training signals into a dictionary and a coefficient matrix constitutes a local minimum. We also analyse statistically when these conditions are fulfilled with high probability.

In this paper, we consider the dynamic Voronoi diagram problem. In this problem, a given set of planar points are moving and our objective is to find the Voronoi diagram of these moving points at any time t. A preprocessing algorithm and a query processing algorithm are presented in this paper. Assume that the points are in k-motion, and it takes O(k) time to find the roots of a polynomial with degree O(k). The preprocessing algorithm uses \(O(k^2 n^3 logn\cdot2^{O(\alpha (n)^{5k + 1} )} )\) time to process moving functions of given points, and uses \(O(k^2 n^3 \cdot2^{O(\alpha (n)^{5k + 1} )} )\) space to store the preprocessing result where α(n) is the functional inverse of Ackermann's function. The query processing algorithm is designed to report the Voronoi diagram of these points for a query time t. It takes O(n) time which is optimal.KeywordsVoronoi DiagramDelaunay TriangulationQuery TimePoint PairPerpendicular BisectorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Regular triangulations of dynamic sets of points

This work has three main parts: the bisectors of curves in the plane, the bisectors of surfaces in \(\mathbb{R}^{3}\), and the Voronoï diagram of a finite family of parallel half-lines in \(\mathbb{R}^{3}\). These subjects are closely related, and have applications in CAD/CAGD and Computational Geometry.We present a new approach to determine, using Cramer’s rule and suitable elimination steps, an exact algebraic parameterization (rational or non rational) of the bisector curve of two given planar rational curves. The approach is, then, generalized by using the generalized Cramer’s rule to determine an exact algebraic parameterization of the bisector surface of two low degree rational surfaces. We apply the method to obtain parametrizations of the bisector of two rational plane curves, when one of them is a circle or a line. On the other hand, we show how to easily obtain parametrizations of the bisector of the following pairs of surfaces: plane-quadric, plane-torus, circular cylinder-non developable quadric, circular cylinder-torus, cylinder-cylinder, cylinder-cone and cone-cone. These parametrizations are rational in most cases. In the remaining cases the parametrization involves one square root which is well-suited for a good approximation of the bisector.In addition, a different approach for the bisector curve problem will be presented. This new method uses dynamic color in GeoGebra for the computation of a geometric and numerical characterization of the bisector of two planar geometric objects (two curves, or a curve and a point). Even if it does not provide an algebraic representation, the method could lead to the computation of an approximate representation of the bisector curve.The Voronoï diagram (VD) is a fundamental data structure in computational geometry with various applications in theoretical and practical areas. We consider the VD of a set of parallel half-lines constrained to a compact domain \(\mathcal{D}_{0} \subset \mathbb{R}^{3}\), with respect to the Euclidean distance. This new kind of VD can be applied to the solution of some problems in the drilling industry, such as hydraulic or mining. We present an efficient approximate algorithm for computing such VD, using a box subdivision process, which produces a mesh representing the topology of the VD.KeywordsComputational GeometryInterval ArithmeticRational CurfRational SurfaceCompact DomainThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Given a triangulation of a point set in the plane, a flip deletes an edge e whose removal leaves a convex quadrilateral, and replaces e by the opposite diagonal of the quadrilateral. It is well known that any triangulation of a point set can be reconfigured to any other triangulation by some sequence of flips. We explore this question in the setting where each edge of a triangulation has a label, and a flip transfers the label of the removed edge to the new edge. It is not true that every labelled triangulation of a point set can be reconfigured to every other labelled triangulation via a sequence of flips, but we characterize when this is possible. There is an obvious necessary condition: for each label l, if edge e has label l in the first triangulation and edge f has label l in the second triangulation, then there must be some sequence of flips that moves label l from e to f, ignoring all other labels. Bose, Lubiw, Pathak and Verdonschot formulated the Orbit Conjecture, which states that this necessary condition is also sufficient, i.e. that all labels can be simultaneously mapped to their destination if and only if each label individually can be mapped to its destination. We prove this conjecture. Furthermore, we give a polynomial-time algorithm to find a sequence of flips to reconfigure one labelled triangulation to another, if such a sequence exists, and we prove an upper bound of O(n^7) on the length of the flip sequence. Our proof uses the topological result that the sets of pairwise non-crossing edges on a planar point set form a simplicial complex that is homeomorphic to a high-dimensional ball (this follows from a result of Orden and Santos; we give a different proof based on a shelling argument). The dual cell complex of this simplicial ball, called the flip complex, has the usual flip graph as its 1-skeleton. We use properties of the 2-skeleton of the flip complex to prove the Orbit Conjecture.

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.