Towards a theory of automated elliptic mesh generation

The theory of elliptic mesh generation is reviewed and the fundamental problem of constructing computational space is discussed. It is argued that the construction of computational space is an NP-Complete problem and therefore requires a nonstandard approach for its solution. This leads to the development of graph-theoretic, combinatorial optimization and integer programming algorithms. Methods for the construction of two dimensional computational space are presented

A Data Structure for Spatio-Temporal Databases

The advantages and applications of spatial mechanisms are well documented; however, there are very few being designed. The principal hinderance to the design of spatial mechanisms is the great difficulty involved in specifying spatial problems and in interpreting spatial solutions. Similarly, the development of spatial codes to implement these techniques is held back by the lack of means to easily visualize and verify solutions, particularly in the realm of relational databases. If spatial mechanisms are to be successful, the designer must be able to synthesize, analyse and evaluate, as well as load and extract information, using a single code representing a spatial structure. This entails the implementation of spatial relationships involving spatial data structures. It is with this in mind that the Canadian Hydrographic Service database group embarked on the development of a new type of spatial database structure based on the quadtree concept

Parallel rendering algorithms for distributed-memory multicomputers

Ankara : Department of Computer Engineering and Information Science and the Institute of Engineering and Science of Bilkent University, 1997.Thesis (Ph. D.) -- Bilkent University, 1997.Includes bibliographical references leaves 166-176.Kurç, Tahsin MertefePh.D

Diamond-based models for scientific visualization

Hierarchical spatial decompositions are a basic modeling tool in a variety of application domains including scientific visualization, finite element analysis and shape modeling and analysis. A popular class of such approaches is based on the regular simplex bisection operator, which bisects simplices (e.g. line segments, triangles, tetrahedra) along the midpoint of a predetermined edge. Regular simplex bisection produces adaptive simplicial meshes of high geometric quality, while simplifying the extraction of crack-free, or conforming, approximations to the original dataset. Efficient multiresolution representations for such models have been achieved in 2D and 3D by clustering sets of simplices sharing the same bisection edge into structures called diamonds. In this thesis, we introduce several diamond-based approaches for scientific visualization. We first formalize the notion of diamonds in arbitrary dimensions in terms of two related simplicial decompositions of hypercubes. This enables us to enumerate the vertices, simplices, parents and children of a diamond. In particular, we identify the number of simplices involved in conforming updates to be factorial in the dimension and group these into a linear number of subclusters of simplices that are generated simultaneously. The latter form the basis for a compact pointerless representation for conforming meshes generated by regular simplex bisection and for efficiently navigating the topological connectivity of these meshes. Secondly, we introduce the supercube as a high-level primitive on such nested meshes based on the atomic units within the underlying triangulation grid. We propose the use of supercubes to associate information with coherent subsets of the full hierarchy and demonstrate the effectiveness of such a representation for modeling multiresolution terrain and volumetric datasets. Next, we introduce Isodiamond Hierarchies, a general framework for spatial access structures on a hierarchy of diamonds that exploits the implicit hierarchical and geometric relationships of the diamond model. We use an isodiamond hierarchy to encode irregular updates to a multiresolution isosurface or interval volume in terms of regular updates to diamonds. Finally, we consider nested hypercubic meshes, such as quadtrees, octrees and their higher dimensional analogues, through the lens of diamond hierarchies. This allows us to determine the relationships involved in generating balanced hypercubic meshes and to propose a compact pointerless representation of such meshes. We also provide a local diamond-based triangulation algorithm to generate high-quality conforming simplicial meshes

Time-Optimal Algorithms on Meshes With Multiple Broadcasting

The mesh-connected computer architecture has emerged as a natural choice for solving a large number of computational tasks in image processing, computational geometry, and computer vision. However, due to its large communication diameter, the mesh tends to be slow when it comes to handling data transfer operations over long distances. In an attempt to overcome this problem, mesh-connected computers have recently been augmented by the addition of various types of bus systems. One such system known as the mesh with multiple broadcasting involves enhancing the mesh architecture by the addition of row and column buses. The mesh with multiple broadcasting has proven to be feasible to implement in VLSI, and is used in the DAP family of computers. In recent years, efficient algorithms to solve a number of computational problems on meshes with multiple broadcasting have been proposed in the literature. The problems considered in this thesis are semigroup computations, sorting, multiple search, various convexity-related problems, and some tree problems. Based on the size of the input data for the problem under consideration, existing results can be broadly classified into sparse and dense. Specifically, for a given √n x √n mesh with multiple broadcasting, we refer to problems involving $m \in O(\sqrt{n}$) items as sparse, while the case £ O(n) will be referred to as dense. Finally, the case corresponding to 2 ≤ m ≤ n is be termed general. The motivation behind the current work is twofold. First, time-optimal solutions are proposed for the problems listed above. Secondly, an attempt is made to remove the artificial limitation of problems studied to sparse and dense cases. To establish the time-optimality of the algorithms presented in this work, we use some existing lower bound techniques along with new ones that we develop. We solve the semigroup computation problem for the general case and present a novel lower bound argument. We solve the multiple search problem in the general case and present some surprising applications to computational geometry. In the case of sorting, the general case is defined to be slightly different. For the specified range of the size of input, we present a time and VLSI-optimal algorithm. We also present time lower bound results and matching algorithms for a number of convexity related and tree problems in the sparse case

Computational metric embeddings

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (p. 141-145).We study the problem of computing a low-distortion embedding between two metric spaces. More precisely given an input metric space M we are interested in computing in polynomial time an embedding into a host space M' with minimum multiplicative distortion. This problem arises naturally in many applications, including geometric optimization, visualization, multi-dimensional scaling, network spanners, and the computation of phylogenetic trees. We focus on the case where the host space is either a euclidean space of constant dimension such as the line and the plane, or a graph metric of simple topological structure such as a tree. For Euclidean spaces, we present the following upper bounds. We give an approximation algorithm that, given a metric space that embeds into R1 with distortion c, computes an embedding with distortion c(1) [delta]3/4 (A denotes the ratio of the maximum over the minimum distance). For higher-dimensional spaces, we obtain an algorithm which, for any fixed d > 2, given an ultrametric that embeds into Rd with distortion c, computes an embedding with distortion co(1). We also present an algorithm achieving distortion c logo(1) [delta] for the same problem. We complement the above upper bounds by proving hardness of computing optimal, or near-optimal embeddings. When the input space is an ultrametric, we show that it is NP-hard to compute an optimal embedding into R2 under the ... norm. Moreover, we prove that for any fixed d > 2, it is NP-hard to approximate the minimum distortion embedding of an n-point metric space into Rd within a factor of Q(n1/(17d)). Finally, we consider the problem of embedding into tree metrics. We give a 0(1)approximation algorithm for the case where the input is the shortest-path metric of an unweighted graph.(cont.) For general metric spaces, we present an algorithm which, given an n-point metric that embeds into a tree with distortion c, computes an embedding with distortion (clog n)o ... . By composing this algorithm with an algorithm for embedding trees into R1, we obtain an improved algorithm for embedding general metric spaces into R1.by Anastasios Sidiropoulos.Ph.D

Discrete Geometry

[no abstract available

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

Formal verification of analog and mixed signal designs: A survey

Analog and mixed signal (AMS) designs are an important part of embedded systems that link digital designs to the analog world. Due to challenges associated with its verification process, AMS designs require a considerable portion of the total design cycle time. In contrast to digital designs, the verification of AMS systems is a challenging task that requires lots of expertise and deep understanding of their behavior. Researchers started lately studying the applicability of formal methods for the verification of AMS systems as a way to tackle the limitations of conventional verification methods like simulation. This paper surveys research activities in the formal verification of AMS designs as well as compares the different proposed approaches

LIPIcs, Volume 258, SoCG 2023, Complete Volume

LIPIcs, Volume 258, SoCG 2023, Complete Volum