Visibility-Related Problems on Parallel Computational Models

Visibility-related problems find applications in seemingly unrelated and diverse fields such as computer graphics, scene analysis, robotics and VLSI design. While there are common threads running through these problems, most existing solutions do not exploit these commonalities. With this in mind, this thesis identifies these common threads and provides a unified approach to solve these problems and develops solutions that can be viewed as template algorithms for an abstract computational model. A template algorithm provides an architecture independent solution for a problem, from which solutions can be generated for diverse computational models. In particular, the template algorithms presented in this work lead to optimal solutions to various visibility-related problems on fine-grain mesh connected computers such as meshes with multiple broadcasting and reconfigurable meshes, and also on coarse-grain multicomputers. Visibility-related problems studied in this thesis can be broadly classified into Object Visibility and Triangulation problems. To demonstrate the practical relevance of these algorithms, two of the fundamental template algorithms identified as powerful tools in almost every algorithm designed in this work were implemented on an IBM-SP2. The code was developed in the C language, using MPI, and can easily be ported to many commercially available parallel computers

Shortest Path in a Polygon using Sublinear Space

\renewcommand{\Re}{{\rm I\!\hspace{-0.025em} R}} \newcommand{\SetX}{\mathsf{X}} \newcommand{\VorX}[1]{\mathcal{V} \pth{#1}} \newcommand{\Polygon}{\mathsf{P}} \newcommand{\Space}{\overline{\mathsf{m}}} \newcommand{\pth}[2][\!]{#1\left({#2}\right)} We resolve an open problem due to Tetsuo Asano, showing how to compute the shortest path in a polygon, given in a read only memory, using sublinear space and subquadratic time. Specifically, given a simple polygon \Polygon with $n$ vertices in a read only memory, and additional working memory of size \Space, the new algorithm computes the shortest path (in \Polygon) in O( n^2 /\, \Space ) expected time. This requires several new tools, which we believe to be of independent interest

Visual analytics of geo-related multidimensional data

In recent years, both the volume and the availability of urban data related to various social issues, such as real estate, crime and population are rapidly increasing. Analysing such urban data can help the government make evidence-based decisions leading to better-informed policies; the citizens can also benefit in many scenarios such as home-seeking. However, the analytic design process can be challenging since (i) the urban data often has multiple attributes (e.g., the distance to supermarket, the distance to work, schools zone in real estate data) that are highly related to geography; and (ii) users might have various analysis/exploration tasks that are hard to define (e.g., different home-buyers might have requirements for housing properties and many of them might not know what they want before they understand the local real estate market). In this thesis, we use visual analytics techniques to study such geo-related multidimensional urban data and answer the following research questions. In the first research question, we propose a visual analytics framework/system for geo-related multidimensional data. Since visual analytics and visualization designs are highly domain-specific, we use the real estate domain as an example to study the problem. Specifically, we first propose a problem abstraction to satisfy the requirements from users (e.g., home buyers, investors). Second, we collect, integrate and clean the last ten year's real estate sold records in Australia as well as their location-related education, facility and transportation profiles, to generate a real multi-dimensional data repository. Third, we propose an interactive visual analytic procedure to help less informed users gradually learn about the local real estate market, upon which users exploit this learned knowledge to specify their personalized requirements in property seeking. Fourth, we propose a series of designs to visualize properties/suburbs in different dimensions and different granularity. Finally, we implement a system prototype for public access (http://115.146.89.158), and present case studies based on real-world datasets and real scenario to demonstrate the usefulness and effectiveness of our system. Our second research question extends the first one and studies the scalability problem to support cluster-based visualization for large-scale geo-related multidimensional data. Particularly, we first propose a design space for cluster-based geographic visualization. To calculate the geographic boundary of each cluster, we propose a concave hull algorithm which can avoid complex shapes, large empty area inside the boundary and overlaps among different clusters. Supported by the concave hull algorithm, we design a cluster-based data structure named ConcaveCubes to efficiently support interactive response to users' visual exploration on large-scale geo-related multidimensional data. Finally, we build a demo system (http://115.146.89.158/ConcaveCubes) to demonstrate the cluster-based geographic visualization, and present extensive experiments using real-world datasets and compare ConcaveCubes with state-of-the-art cube-based structures to verify the efficiency and effectiveness of ConcaveCubes. The last research question studies the problem related to visual analytics of urban areas of interest (AOIs), where we visualize geographic points that satisfy the user query as a limited number of regions (AOIs) instead of a large number of individual points (POIs). After proposing a design space for AOI visualization, we design a parameter-free footprint method named AOI-shapes to effectively capture the region of an AOI based on POIs that satisfy the user query and those that do not. We also propose two incremental methods which generate the AOI-shapes by reusing previous calculations as per users' update of their AOI query. Finally, we implement an online demo (http://www.aoishapes.com) and conduct extensive experiments to demonstrate the efficiency and effectiveness of the proposed AOI-shapes

Reconstruction of Orthogonal Polyhedra

In this thesis I study reconstruction of orthogonal polyhedral surfaces and orthogonal polyhedra from partial information about their boundaries. There are three main questions for which I provide novel results. The first question is "Given the dual graph, facial angles and edge lengths of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the dihedral angles?" The second question is "Given the dual graph, dihedral angles and edge lengths of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the facial angles?" The third question is "Given the vertex coordinates of an orthogonal polyhedral surface or polyhedron, is it possible to reconstruct the edges and faces, possibly after rotating?" For the first two questions, I show that the answer is "yes" for genus-0 orthogonal polyhedra and polyhedral surfaces under some restrictions, and provide linear time algorithms. For the third question, I provide results and algorithms for orthogonally convex polyhedra. Many related problems are studied as well

A Computational Paradigm on Network-Based Models of Computation

The maturation of computer science has strengthened the need to consolidate isolated algorithms and techniques into general computational paradigms. The main goal of this dissertation is to provide a unifying framework which captures the essence of a number of problems in seemingly unrelated contexts in database design, pattern recognition, image processing, VLSI design, computer vision, and robot navigation. The main contribution of this work is to provide a computational paradigm which involves the unifying framework, referred to as the multiple Query problem, along with a generic solution to the Multiple Query problem. To demonstrate the applicability of the paradigm, a number of problems from different areas of computer science are solved by formulating them in this framework. Also, to show practical relevance, two fundamental problems were implemented in the C language using MPI. The code can be ported onto many commercially available parallel computers; in particular, the code was tested on an IBM-SP2 and on a network of workstations

Discrete Geometry (hybrid meeting)

A number of important recent developments in various branches of discrete geometry were presented at the workshop, which took place in hybrid format due to a pandemic situation. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics, algebraic geometry or functional analysis. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry

Abstracts for the twentyfirst European workshop on Computational geometry, Technische Universiteit Eindhoven, The Netherlands, March 9-11, 2005

This volume contains abstracts of the papers presented at the 21st European Workshop on Computational Geometry, held at TU Eindhoven (the Netherlands) on March 9–11, 2005. There were 53 papers presented at the Workshop, covering a wide range of topics. This record number shows that the field of computational geometry is very much alive in Europe. We wish to thank all the authors who submitted papers and presented their work at the workshop. We believe that this has lead to a collection of very interesting abstracts that are both enjoyable and informative for the reader. Finally, we are grateful to TU Eindhoven for their support in organizing the workshop and to the Netherlands Organisation for Scientific Research (NWO) for sponsoring the workshop

Large bichromatic point sets admit empty monochromatic 4-gons

We consider a variation of a problem stated by Erd˝os and Szekeres in 1935 about the existence of a number fES(k) such that any set S of at least fES(k) points in general position in the plane has a subset of k points that are the vertices of a convex k-gon. In our setting the points of S are colored, and we say that a (not necessarily convex) spanned polygon is monochromatic if all its vertices have the same color. Moreover, a polygon is called empty if it does not contain any points of S in its interior. We show that any bichromatic set of n ≥ 5044 points in R2 in general position determines at least one empty, monochromatic quadrilateral (and thus linearly many).Postprint (published version