An efficient output-sensitive hidden surface removal algorithm and its parallelization

In this paper we present an algorithm for hidden surface removal for a class of polyhedral surfaces which have a property that they can be ordered relatively quickly like the terrain maps. A distinguishing feature of this algorithm is that its running time is sensitive to the actual size of the visible image rather than the total number of intersections in the image plane which can be much larger than the visible image. The time complexity of this algorithm is O((k +nflognloglogn) where n and k are respectively the input and the output sizes. Thus, in a significant number of situations this will be faster than the worst case optimal algorithms which have running time Ω(n 2) irrespective of the output size (where as the output size k is O(n 2) only in the worst case). We also present a parallel algorithm based on a similar approach which runs in time O(log4(n+k)) using O((n + k)/Iog(n+k)) processors in a CREW PRAM model. All our bounds arc obtained using ammortized analysis

GENERAL POLYHEDRAL FINITE ELEMENTS FOR RAPID NONLINEAR ANALYSIS

An analysis system for solid mechanics applications is described in which a new finite element method that can accommodate general polyhedral elements is exploited. The essence of the method is direct polynomial approximation of the shape functions on the physical element, without transformation to a canonical element. The main motive is elimination of the requirement that all elements be similar to a canonical element via the usual isoparametric mapping. It is this topological restriction that largely drives the design of mesh-generation algorithms, and ultimately leads to the considerable human effort required to perform complex analyses. An integrated analysis system is described in which the flexibility of the polyhedral element method is leveraged via a robust computational geometry processor.T he role of the latter is to perform rapid Boolean intersection operations between hex meshes and surface representations of the body to be analyzed. A typical procedure is to create a space-filling structured hex mesh that contains the body, and then extract a polyhedral mesh of the body by intersecting the hex mesh and the body's surface. The result is a mesh that is directly usable in the polyhedral finite element method. Some example applications are: 1) simulation on very complex geometries; 2) rapid geometry modification and re-analysis; and 3) analysis of material-removal process steps following deformation processing. This last class of problems is particularly challenging for the conventional FE methodology, because the element boundaries are, in general, not aligned with the †Corresponding author

Formulations and algorithms for the maximum area poligonization problem

Orientador: Fábio Luiz UsbertiDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: O problema do caixeiro viajante euclidiano (Traveling Salesman Problem - TSP) do ponto de vista geométrico tem por objetivo encontrar um polígono simples sobre um dado conjunto de vértices cujo perímetro é mínimo. É possível derivar o problema de modo que o objetivo seja encontrar um polígono simples cuja área interna seja máxima: tal problema é conhecido como Poligonização de Área Máxima (Maximum Area Polygonization - MAXAP). O MAXAP é um problema de otimização combinatória NP-difícil com aplicações práticas em segmentos como reconhecimento de padrões, reconstrução de imagens, clusterização e robótica. Este trabalho propõe novas metodologias de solução e formulações matemáticas para o MAXAP, visando a implementação de algoritmos para metodologias exata, aproximada e heurística, bem como um estudo computacional para avaliar o desempenho das metodologias para o conjunto de instâncias desenvolvido. São propostos neste trabalho dois modelos matemáticos de programação linear inteira, duas heurísticas construtivas, uma metaheurística GRASP e uma matheuristic aplicada sobre um dos modelos matemáticos. Experimentos computacionais foram executados para comparar as metodologias propostas entre si e com um algoritmo 1/2-aproximado da literatura. Análises comparativas de desempenho foram realizadas sobre os resultados obtidos mostrando que o GRASP obteve o melhor desempenho no critério de qualidade das soluções. As heurísticas construtivas propostas por sua vez obtiveram um desempenho superior sobre o algoritmo aproximado. Finalmente, os modelos matemáticos propostos mostram a dificuldade de resolver de maneira exata o MAXAP, encontrando soluções ótimas em uma hora somente para as instâncias de 10 pontos, em um conjunto de instâncias de até 100 pontosAbstract: The Traveling Salesman Problem (TSP) from a geometric point of view aims to find a simple polygon with minimum perimeter. It is possible to derive the problem so that the objective is to find a simple polygon whose enclosed area is maximum, such problem is known as Maximum Area Polygonization (MAXAP). The MAXAP is an NP-hard combinatorial optimization problem with practical applications in segments such as pattern recognition, image reconstruction, clustering and robotics. This work proposes new solution methodologies and mathematical formulations for MAXAP, aiming the implementation of algorithms for exact, approximate and heuristic solutions, as well as a computational study to evaluate the performance of the methodologies for a benchmark set of instances. Two mathematical models based on integer linear programming are proposed. In addition, two constructive heuristics, a GRASP metaheuristic, and a matheuristic are proposed for the solution of larger instances. Computacional experiments were conducted to compare the proposed methodologies among themselves and a 1/2-approximation algorithm from literature. Comparative perfomance analysis were made on the results showing that the GRASP outperformed all other approaches with respect to solution quality. The constructive heuristics, on the other hand, outperformed the literature 1/2-approximation algorithm. Finally, the proposed mathematical models have shown the hardness of exact solution for the MAXAP, finding optimal solutions in one hour only for the 10-vertices instances in a set of instances with 10, 25, 50 and 100 verticesMestradoCiência da ComputaçãoMestre em Ciência da ComputaçãoCAPE

Approximate Polytope Membership Queries

International audienceIn the polytope membership problem, a convex polytope K in R d is given, and the objective is to preprocess K into a data structure so that, given any query point q ∈ R d , it is possible to determine efficiently whether q ∈ K. We consider this problem in an approximate setting. Given an approximation parameter ε, the query can be answered either way if the distance from q to K's boundary is at most ε times K's diameter. We assume that the dimension d is fixed, and K is presented as the intersection of n halfspaces. Previous solutions to approximate polytope membership were based on straightforward applications of classic polytope approximation techniques by Dudley (1974) and Bentley et al. (1982). The former is optimal in the worst-case with respect to space, and the latter is optimal with respect to query time. We present four main results. First, we show how to combine the two above techniques to obtain a simple space-time trade-off. Second, we present an algorithm that dramatically improves this trade-off. In particular, for any constant α ≥ 4, this data structure achieves query time roughly O 1/ε (d−1)/α and space roughly O 1/ε (d−1)(1−Ω(log α)/α). We do not know whether this space bound is tight, but our third result shows that there is a convex body such that our algorithm achieves a space of at least Ω 1/ε (d−1)(1−O(√ α)/α. Our fourth result shows that it is possible to reduce approximate Euclidean nearest neighbor searching to approximate polytope membership queries. Combined with the above results, this provides significant improvements to the best known space-time trade-offs for approximate nearest neighbor searching in R d. For example, we show that it is possible to achieve a query time of roughly O(log n + 1/ε d/4) with space roughly O(n/ε d/4), thus reducing by half the exponent in the space bound