Optimal Point Placement for Mesh Smoothing

We study the problem of moving a vertex in an unstructured mesh of triangular, quadrilateral, or tetrahedral elements to optimize the shapes of adjacent elements. We show that many such problems can be solved in linear time using generalized linear programming. We also give efficient algorithms for some mesh smoothing problems that do not fit into the generalized linear programming paradigm.Comment: 12 pages, 3 figures. A preliminary version of this paper was presented at the 8th ACM/SIAM Symp. on Discrete Algorithms (SODA '97). This is the final version, and will appear in a special issue of J. Algorithms for papers from SODA '9

On a Linear Program for Minimum-Weight Triangulation

Minimum-weight triangulation (MWT) is NP-hard. It has a polynomial-time constant-factor approximation algorithm, and a variety of effective polynomial- time heuristics that, for many instances, can find the exact MWT. Linear programs (LPs) for MWT are well-studied, but previously no connection was known between any LP and any approximation algorithm or heuristic for MWT. Here we show the first such connections: for an LP formulation due to Dantzig et al. (1985): (i) the integrality gap is bounded by a constant; (ii) given any instance, if the aforementioned heuristics find the MWT, then so does the LP.Comment: To appear in SICOMP. Extended abstract appeared in SODA 201

On a linear program for minimum-weight triangulation

Minimum-weight triangulation (MWT) is NP-hard. It has a polynomial-time constant-factor approximation algorithm, and a variety of effective polynomial-time heuristics that, for many instances, can find the exact MWT. Linear programs (LPs) for MWT are well-studied, but previously no connection was known between any LP and any approximation algorithm or heuristic for MWT. Here we show the first such connections: For an LP formulation due to Dantzig, Hoffman, and Hu [Math. Programming, 31 (1985), pp. 1-14], (i) the integrality gap is constant, and (ii) given any instance, if the aforementioned heuristics find the MWT, then so does the LP. © 2014 Society for Industrial and Applied Mathematics

Globally optimal triangulations of minimum weight using Ant Colony Optimization metaheuristic

Globally optimal triangulations are difficult to be found by deterministic methods as, for most type of criteria, no polynomial algorithm is known. In this work, we consider the Minimum Weight Triangulation (MWT) problem of a given set of n points in the plane. Our aim is to show how the Ant Colony Optimization (ACO) metaheuristic can be used to search for globally optimal triangulations of minimum weight. We present an experimental study for a set of instances for MWT problem. We create these instances since no reference to benchmarks for this problem were found in the literature. We assess through the experimental evaluation the applicability of the ACO metaheuristic for MWT problem.Facultad de Informátic

Studies of several tetrahedralization problems

The main purpose of decomposing an object into simpler components is to simplify a problem involving the complex object into a number of subproblems having simpler components. In particular, a tetrahedralization is a partition of the input domain in R3 into a number of tetrahedra that meet only at shared faces. Tetrahedralizations have applications in the finite element method, mesh generation, computer graphics, and robotics. This thesis investigates four problems in tetrahedralizations and triangulations. The first problem is on the computational complexity of tetrahedralization detections. We present an O(nm log n) algorithm to determine whether a set of line segments .C is the edge set of a tetrahedralization, where m is the number of segments and n is the number of endpoints in .C. We show that it is NP-complete to decide whether .C contains the edge set of a tetrahedralization. We also show that it is NP-complete to decide whether .C is tetrahedralizable. The second problem is on minimal tetrahedralizations. After deriving some properties of the graph of polyhedra, we identify a class of polyhedra and show that this class of polyhedra can be minimally tetrahedralized in O(n²) time. The third problem is on the tetrahedralization of two nested convex polyhedra. We give a method to tetrahedralize the region between two nested convex polyhedra into a linear number of tetrahedra without introducing Steiner points. This result answers an open problem raised by Bern [16]. The fourth problem is on the lower bound for β-skeletons belonging to minimum weight triangulations. We prove a lower bound on β (β = [one sixth times the square root of two times the square root of 3] + 45 such that if β is less than this value, the β-skeleton of a point set may not always be a subgraph of the minimum weight triangulation of this point set. This result settles Keil's conjecture [62]

Classificação e partição de polígonos simples

Mestrado em Matemática - EnsinoEsta dissertação tem como objectivo fazer um estudo sobre polígonos simples, nomeadamente no que concerne à sua classificação e partição. Começa-se por apresentar várias classes de polígonos simples fazendo depois uma classificação hierárquica. São apresentados alguns exemplos de polígonos simples segundo algumas características específicas. Posteriormente abordase o tema da partição clássica de polígonos simples. Faz-se uma resenha histórica sobre a evolução da complexidade da triangulação de polígonos simples, apresentam-se os algoritmos mais marcantes deste tipo de partição e mostra-se como, a partir de polígonos simples triangulados, se pode obter uma quadrangulação. Faz-se, também, uma abordagem a uma partição não clássica, como é o caso da pseudo-triangulação. Por fim, apresentam-se alguns problemas que ainda permanecem em aberto.The goal of this dissertation is to study simple polygons, namely concerning their classification and partition. We start by presenting several classes of simple polygons, performing next a sorted classification. Some examples of simple polygons are presented according to some specific characteristics. The classical partition of simple polygons theme is discussed next. We make an historical draft on the evolution of the triangulation complexity of simple polygons, the fundamental algorithms of this type of partition are described, and its shown how, starting with simple triangulated polygons, we can obtain a quadrangulation. An approach to non-classic partitions is done, e.g. the pseudo-triangulation. At last, some problems that remain unsolved are presented