Graph algorithms are concerned with the algorithmic aspects of solving graph problems. The problems are motivated from and have application to diverse areas of computer science, engineering and other disciplines. Problems arising from these areas of application are good candidates for parallelization since they often have both intense computational needs and stringent response time requirements. Motivated by these concerns, this thesis investigates parallel algorithms for these kinds of graph problems that have at least one of the following properties: the problems involve some type of dynamic updates; the sparsification technique is applicable; or the problems are closely related to communications network issues. The models of parallel computation used in our studies are the Parallel Random Access Machine (PRAM) model and the practical interconnection network models such as meshes and hypercubes. ¶ Consider a communications network which can be represented by a graph G = (V;E), where V is a set of sites (processors), and E is a set of links which are used to connect the sites (processors). In some cases, we also assign weights and/or directions to the edges in E. Associated with this network, there are many problems such as (i) whether the network is k-edge (k-vertex) connected withfixed k; (ii) whether there are k-edge (k-vertex) disjoint paths between u and v for a pair of given vertices u and v after the network is dynamically updated by adding and/or deleting an edge etc; (iii) whether the sites in the network can communicate with each other when some sites and links fail; (iv) identifying the first k edges in the network whose deletion will result in the maximum increase in the routing cost in the resulting network for fixed k; (v) how to augment the network at optimal cost with a given feasible set of weighted edges such that the augmented network is k-edge (k-vertex) connected; (vi) how to route messages through the network efficiently. In this thesis we answer the problems mentioned above by presenting efficient parallel algorithms to solve them. As far as we know, most of the proposed algorithms are the first ones in the parallel setting. ¶ Even though most of the problems concerned in this thesis are related to communications networks, we also study the classic edge-coloring problem. The outstanding difficulty to solve this problem in parallel is that we do not yet know whether or not it is in NC. In this thesis we present an improved parallel algorithm for the problem which needs [bigcircle]([bigtriangleup][superscript 4.5]log [superscript 3] [bigtriangleup] log n + [bigtriangleup][superscript 4] log [superscript 4] n) time using [bigcircle](n[superscript 2][bigtriangleup] + n[bigtriangleup][superscript 3]) processors, where n is the number of vertices and [bigtriangleup] is the maximum vertex degree. Compared with a previously known result on the same model, we improved by an [bigcircle]([bigtriangleup][superscript 1.5]) factor in time. The non-trivial part is to reduce this problem to the edge-coloring update problem. We also generalize this problem to the approximate edge-coloring problem by giving a faster parallel algorithm for the latter case. ¶ Throughout the design and analysis of parallel graph algorithms, we also find a technique called the sparsification technique is very powerful in the design of efficient sequential and parallel algorithms on dense undirected graphs. We believe that this technique may be useful in its own right for guiding the design of efficient sequential and parallel algorithms for problems in other areas as well as in graph theory.

The first ocean general circulation models developed in the late sixties were based on finite differences schemes on structured grids. Many improvements in the fields of engineering have been achieved since three decades with the developments of new numerical methods based on unstructured meshes. Some components of the first models may now seem out of date and new second generation models are therefore under study, with the aim of taking advantage of the potential of modern numerical techniques such as finite elements. In particular, unstructured meshes are believed to be more efficient to resolve the large range of time and space scales present in the ocean. Besides the classical continuous finite element or finite volume methods, another popular new trend in engineering applications is the Discontinuous Galerkin (DG) method, i.e. discontinuous finite elements presenting many interesting numerical properties in terms of dispersion and dissipation, errors convergence rates, advection schemes, mesh adaptation, etc. The method is especially efficient at high polynomial orders. The motivation for this PhD research is therefore to investigate the use of the high-order DG method for geophysical flow modeling. A first part of the thesis is devoted to the mesh adaptation using the DG method. The inter-element jumps of the fields are used as error estimators. New mesh size fields or polynomial orders are then derived and local h- or p-adaptation is performed. The technique is applied to standard benchmarks and computations in more realistic domains as the Gulf of Mexico. A second part deals with the use of the high order DG method with high-order representation of geometrical features. On one hand, a method is proposed to deal with complex representations of the coastlines. Computations are performed using high-order mappings around the Rattray island, located in the Great Barier Reef. Numerical results are then compared to in-situ measurements. On the other hand, a new method is proposed to deal with curved manifolds in order to represents oceanic or atmospheric flows on the sphere. The approach is based on the use of a local high-order non-orthogonal basis, and is equivalent to the use of vectorial shape and test functions to represent the vectorial conservation laws on the manifold's surface. A method is finally proposed to analyze the dispersion and dissipation properties of any numerical scheme on any kind of grid, possibly unstructured. The DG method is then compared to other techniques as the mixed non-conforming linear elements, and the impact of unstructured meshes is studied.

Los taludes recién construidos presentan un aspecto adecuado y una superficie regular y estable, pero por diversos agentes se erosionan rápidamente. Una técnica a utilizar para la estabilización y control de la erosión es el empleo de mallas arriostradas al terreno mediante anclajes. Esto facilita la revegetación, la integración paisajística y un bajo impacto medioambiental. Los resultados han sido dos modelos físico-matemáticos para el diseño de dos sistemas flexibles antierosión y de estabilización superficial de desmontes. En el primero, la membrana se encuentra arriostrada a la ladera mediante anclajes en líneas horizontales y al tresbolillo (Modelo Puntual). En el segundo, la malla transmite esfuerzos a elementos de arriostre y estos a la cabeza de los anclajes (Modelo Bidireccional). Para validar los modelos, se diseñaron tres instalaciones para las membranas flexibles, objeto de patente. Para este tipo de sistemas está indicado el uso de membranas flexibles anisótropas. Estos sistemas evitan la erosión del talud, recogen presiones ejercidas por este y las transmiten a la parte estable del anclaje. / Newly built slopes show a good aspect and a regular and stable surface, but for different agents quickly erode. One technique to use for the stabilization and control of erosion is the use of meshes braced to the ground through anchors. This facilitates revegetation, landscape integration, and a low environmental impact. The resuls of the study are two physical-mathematic models for the design of two flexible anti-erosion systems and for the excavation surface stabilization. In the first, the mesh is braced to the slope through anchors in horizontal lines and staggered (Punctual Model). In the second, the mesh transmits stresses to bracing elements and these to the anchors heads (Bidirectional Model). To validate the models, three installations for the flexible membranes were designed, the object of patents. For this type of systems, the use of flexible anisotropic membranes is indicated. These systems avoid slope erosion, gather the pressures exerted by it and transmit them to the anchor stable part.

The paper presents a method for generating curved surfaces of 3D finite element meshes by mesh refinement starting with a very coarse grid. This is useful for parallel implementations where the finest meshes should be computed and not read from large files. The paper deals with simple geometries as sphere, cylinder, cone. But the method may be extended to more complicated geometries. (with 45 figures)

In this work we study the framework of mathematical morphology on simplicial complex spaces. Simplicial complexes are a versatile and widely used structure to represent multidimensional data, such as meshes, that are tridimensional complexes, or graphs, that can be interpreted as bidimensional complexes. Mathematical morphology is one of the most powerful frameworks for image processing, including the processing of digital structures, and is heavily used for many applications. However, mathematical morphology operators on simplicial complex spaces is not a concept fully developped in the literature. In this work, we review some classical operators from simplicial complexes under the light of mathematical morphology, to show that they are morphology operators. We define some basic lattices and operators acting on these lattices: dilations, erosions, openings, closings and alternating sequential filters, including their extension to weighted simplexes. However, the main contributions of this work are what we called dimensional operators, small, versatile operators that can be used to define new operators on simplicial complexes, while mantaining properties from mathematical morphology. These operators can also be used to express virtually any operator from the literature. We illustrate all the defined operators and compare the alternating sequential filters against filters defined in the literature, where our filters show better results for removal of small, intense, noise from binary images

In scientific computing, it is very common to visualize the approximate solution obtained by a numerical PDE solver by drawing surface or contour plots of all or some components of the associated approximate solutions. These plots are used to investigate the behavior of the solution and to display important properties or characteristics of the approximate solutions. In this thesis, we consider techniques for drawing such contour plots for the solution of two and three dimensional PDEs. We first present three fast contouring algorithms in two dimensions over an underlying unstructured mesh. Unlike standard contouring algorithms, our algorithms do not require a fine structured approximation. We assume that the underlying PDE solver generates approximations at some scattered data points in the domain of interest. We then generate a piecewise cubic polynomial interpolant (PCI) which approximates the solution of a PDE at off-mesh points based on the DEI (Differential Equation Interpolant) approach. The DEI approach assumes that accurate approximations to the solution and first-order derivatives exist at a set of discrete mesh points. The extra information required to uniquely define the associated piecewise polynomial is determined based on almost satisfying the PDE at a set of collocation points. In the process of generating contour plots, the PCI is used whenever we need an accurate approximation at a point inside the domain. The direct extension of the both DEI-based interpolant and the contouring algorithm to three dimensions is also investigated. The use of the DEI-based interpolant we introduce for visualization can also be used to develop effective Adaptive Mesh Refinement (AMR) techniques and global error estimates. In particular, we introduce and investigate four AMR techniques along with a hybrid mesh refinement technique. Our interest is in investigating how well such a `generic' mesh selection strategy, based on properties of the problem alone, can perform compared with a special-purpose strategy that is designed for a specific PDE method. We also introduce an \`{a} posteriori global error estimator by introducing the solution of a companion PDE defined in terms of the associated PCI.

In scientific computing, it is very common to visualize the approximate solution obtained by a numerical PDE solver by drawing surface or contour plots of all or some components of the associated approximate solutions. These plots are used to investigate the behavior of the solution and to display important properties or characteristics of the approximate solutions. In this thesis, we consider techniques for drawing such contour plots for the solution of two and three dimensional PDEs. We first present three fast contouring algorithms in two dimensions over an underlying unstructured mesh. Unlike standard contouring algorithms, our algorithms do not require a fine structured approximation. We assume that the underlying PDE solver generates approximations at some scattered data points in the domain of interest. We then generate a piecewise cubic polynomial interpolant (PCI) which approximates the solution of a PDE at off-mesh points based on the DEI (Differential Equation Interpolant) approach. The DEI approach assumes that accurate approximations to the solution and first-order derivatives exist at a set of discrete mesh points. The extra information required to uniquely define the associated piecewise polynomial is determined based on almost satisfying the PDE at a set of collocation points. In the process of generating contour plots, the PCI is used whenever we need an accurate approximation at a point inside the domain. The direct extension of the both DEI-based interpolant and the contouring algorithm to three dimensions is also investigated. The use of the DEI-based interpolant we introduce for visualization can also be used to develop effective Adaptive Mesh Refinement (AMR) techniques and global error estimates. In particular, we introduce and investigate four AMR techniques along with a hybrid mesh refinement technique. Our interest is in investigating how well such a `generic' mesh selection strategy, based on properties of the problem alone, can perform compared with a special-purpose strategy that is designed for a specific PDE method. We also introduce an \`{a} posteriori global error estimator by introducing the solution of a companion PDE defined in terms of the associated PCI.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O presente trabalho propõe analisar metodologias para o cálculo do gradiente em malhas não-estruturadas do tipo Voronoi que são utilizadas no método de Volumes Finitos. Quatro metodologias para o cálculo do gradiente são testadas e comparadas com soluções analíticas. As técnicas utilizadas são: Método do Balanço de Forças, Método do Mínimo Resíduo Quadrático, Método da Média dos Gradientes Projetos e Método da Média dos Gradientes Projetados Corrigidos. Uma análise por série de Taylor também foi feita, e as equações analíticas comparadas com resultados numéricos. Os testes são realizados em malhas cartesianas e malhas triangulares, que em um trabalho anterior apresentaram alguns resultados inconsistentes. A influência do ponto gerador e do ângulo de rotação é analisada. É verificado que a posição do ponto gerador e a metodologia utilizada em cada malha influencia no cálculo do gradiente. Dependendo da malha e da metodologia utilizada, as equações analíticas indicaram que existem erros associados, que prejudicam o cálculo do gradiente. / Presente work propose examine methodologies for calculate gradient using a non-structured mesh of Voronois type, used on finite volume method. Four methodologies for calculate gradient are tested and compared with analytical solutions. The techniques used are: the Forces Balance Method, Minimum Quadratic Residue Method, Average Projected Gradient Method and the Revised Average Projected Method. An analysis using Taylor series was also made, and the analytical equations compared with numerical results. Tests are performed over Cartesian and triangular meshes, second one which in a previous work showed some inconsistent results. The influence of the gerator point displacement and rotation angle is analyzed. It was found that the position of the generator point and the methodology used influences gradient value. Accordingly to the mesh and the methodology used, analytical equations indicates that there are associated errors, which affect gradient value.

Considerações de informação perceptual têm ganhado espaço rapidamente em pesquisas referentes a representação, análise e exibição de malhas. Estudos com usuários, eye tracking e outras técnicas são capazes de fornecer informações cada vez mais úteis para sistemas voltados a usuário, que formam a maioria das aplicações em computação gráfica. Neste trabalho nós expandimos sobre o conceito de Saliência de Malhas — uma medida automática de importância visual para malhas de triângulos baseada em modelos de atenção humana em baixo nível — melhorando, extendendo e realizando integração com diferentes aplicações. Nós extendemos o conceito de Saliência de Malhas para englobar objetos deformáveis, mostrando como um mapa de saliência em nível de vértice pode ser construído capturando corretamente regiões de alta importância perceptual através de um conjunto de poses ou deformações. Nós definimos saliência multi-pose como um agregado multi-escala de valores de curvatura sobre uma vizinhança localmente estável, em conjunto com deformações desta vizinhança em múltiplas poses. Nós substituímos distância Euclideana por geodésica, assim fornecendo melhores estimativas de vizinhança local. Resultados mostram que saliência multi-pose gera resultados visualmente mais interessantes em simplificações quando comparado à saliência em uma única pose. Nós também aplicamos saliência de malhas ao problema de segmentação e rendering dependente de ponto de vista, introduzindo uma técnica para segmentação que particiona um objeto em um conjunto de clusters, cada um englobando um grupo de características localmente interessantes. Saliência de malhas é incorporada em um framework para clustering propagativo, guiando seleção de pontos de partida para clusters e custos de propagação de faces, levando a uma convergência de clusters ao redor de características perceptualmente importantes. Nós comparamos nossa técnica com diferentes métodos automáticos para segmentação, mostrando que ela fornece segmentação melhor ou comparável sem necessidade de intervenção do usuário. Uma vez que o algoritmo de segmentação proposto é especialmente aplicável a rendering multi-resolução, nós ilustramos uma aplicação do mesmo através de um sistema de rendering baseado em ponto de vista guiado por saliência, alcançando melhorias consideráveis em framerate com muito pouca perda de qualidade visual. / Considerations on perceptual information are quickly gaining importance in mesh representation, analysis and display research. User studies, eye tracking and other techniques are able to provide ever more useful insights for many user-centric systems, which form the bulk of computer graphics applications. In this work we build upon the concept of Mesh Saliency — an automatic measure of visual importance for triangle meshes based on models of low-level human visual attention—improving, extending and integrating it with different applications. We extend the concept of Mesh Saliency to encompass deformable objects, showing how a vertex-level saliency map can be constructed that accurately captures the regions of high perceptual importance over a range of mesh poses or deformations. We define multipose saliency as a multi-scale aggregate of curvature values over a locally stable vertex neighborhood together with deformations over multiple poses. We replace the use of the Euclidean distance by geodesic distance thereby providing superior estimates of the local neighborhood. Results show that multi-pose saliency generates more visually appealing mesh simplifications when compared to a single-pose mesh saliency. We also apply Mesh Saliency to the problem of mesh segmentation and view-dependent rendering, introducing a technique for segmentation that partitions an object into a set of face clusters, each encompassing a group of locally interesting features. Mesh Saliency is incorporated in a propagative mesh clustering framework, guiding cluster seed selection and triangle propagation costs and leading to a convergence of face clusters around perceptually important features. We compare our technique with different fully automatic segmentation algorithms, showing that it provides similar or better segmentation without the need for user input. Since the proposed clustering algorithm is specially suitable for multi-resolution rendering, we illustrate application of our clustering results through a saliency-guided view-dependent rendering system, achieving significant framerate increases with little loss of visual detail.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / O presente trabalho propõe analisar metodologias para o cálculo do gradiente em malhas não-estruturadas do tipo Voronoi que são utilizadas no método de Volumes Finitos. Quatro metodologias para o cálculo do gradiente são testadas e comparadas com soluções analíticas. As técnicas utilizadas são: Método do Balanço de Forças, Método do Mínimo Resíduo Quadrático, Método da Média dos Gradientes Projetos e Método da Média dos Gradientes Projetados Corrigidos. Uma análise por série de Taylor também foi feita, e as equações analíticas comparadas com resultados numéricos. Os testes são realizados em malhas cartesianas e malhas triangulares, que em um trabalho anterior apresentaram alguns resultados inconsistentes. A influência do ponto gerador e do ângulo de rotação é analisada. É verificado que a posição do ponto gerador e a metodologia utilizada em cada malha influencia no cálculo do gradiente. Dependendo da malha e da metodologia utilizada, as equações analíticas indicaram que existem erros associados, que prejudicam o cálculo do gradiente. / Presente work propose examine methodologies for calculate gradient using a non-structured mesh of Voronois type, used on finite volume method. Four methodologies for calculate gradient are tested and compared with analytical solutions. The techniques used are: the Forces Balance Method, Minimum Quadratic Residue Method, Average Projected Gradient Method and the Revised Average Projected Method. An analysis using Taylor series was also made, and the analytical equations compared with numerical results. Tests are performed over Cartesian and triangular meshes, second one which in a previous work showed some inconsistent results. The influence of the gerator point displacement and rotation angle is analyzed. It was found that the position of the generator point and the methodology used influences gradient value. Accordingly to the mesh and the methodology used, analytical equations indicates that there are associated errors, which affect gradient value.