Triangulations intersect nicely

We show that there is a matching between the edges of any two triangulations of a planar point set such that an edge of one triangulation is matched either to the identical edge in the other triangulation or to an edge that crosses it. This theorem also holds for the triangles of the triangulations and in general independence systems. As an application, we give some lower bounds for the minimum-weight triangulation which can be computed in polynomial time by matching and network-flow techniques. We exhibit an easy-to-recognize class of point sets for which the minimum-weight triangulation coincides with the greedy triangulation.     

Efficient geometric multigrid implementation for triangular grids

This paper deals with a stencil-based implementation of a geometric multigrid method on semi-structured triangular grids (triangulations obtained by regular refinement of an irregular coarse triangulation) for linear finite element methods. An efficient and elegant procedure to construct these stencils using a reference stencil associated to a canonical hexagon is proposed. Local Fourier Analysis (LFA) is applied to obtain asymptotic convergence estimates. Numerical experiments are presented to illustrate the efficiency of this geometric multigrid algorithm, which is based on a three-color smoother.     

Uniform Infinite Planar Triangulation and Related Time-Reversed Critical Branching Process

We consider the uniform infinite planar triangulation, which is defined as the weak limit of the uniform distributions on finite triangulations with N triangles as N → ∞. Take the ball of radius R in an infinite triangulation. One of the components of its boundary separates this ball from the infinite part of the triangulation, and we denote its length by ℓ(R). The main question we study is the asymptotic behavior of the sequence ℓ(R), R = 1, 2,..., called the triangulation profile. First, we prove that the ratio ℓ(R)/R² converges to a nondegenerate random variable. Second, we establish a connection between the triangulation profile and a certain time-reversed critical branching process. Finally, we show that there exists a contour of length linear in R that lies outside of the R-ball and separates this ball from the infinite part of the triangulation. Bibliography: 8 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 307, 2004, pp. 141–174.     

A Linear Bound on the Complexity of the Delaunay Triangulation of Points on Polyhedral Surfaces

Delaunay triangulations and Voronoi diagrams have found numerous applications in surface modeling, surface mesh generation, deformable surface modeling and surface reconstruction. Many algorithms in these applications begin by constructing the three-dimensional Delaunay triangulation of a finite set of points scattered over a surface. Their running-time therefore depends on the complexity of the Delaunay triangulation of such point sets. Although the complexity of the Delaunay triangulation of points in R³ may be quadratic in the worst case, we show in this paper that it is only linear when the points are distributed on a fixed set of well-sampled facets of R³ (e.g. the planar polygons in a polyhedron). Our bound is deterministic and the constants are explicitly given.     

Dense Point Sets Have Sparse Delaunay Triangulations or “... But Not Too Nasty”

The spread of a finite set of points is the ratio between the longest and shortest pairwise distances. We prove that the Delaunay triangulation of any set of $n$ points in~$\Real^3$ with spread $\Delta$ has complexity $O(\Delta^3)$. This bound is tight in the worst case for all $\Delta = O(\sqrt{n})$. In particular, the Delaunay triangulation of any dense point set has linear complexity. We also generalize this upper bound to regular triangulations of $k$-ply systems of balls, unions of several dense point sets, and uniform samples of smooth surfaces. On the other hand, for any $n$ and $\Delta = O(n)$, we construct a regular triangulation of complexity $\Omega(n\Delta)$ whose $n$ vertices have spread $\Delta$.     

Diagonal Flips in Labelled Planar Triangulations

 A classical result of Wagner states that any two (unlabelled) planar triangulations with the same number of vertices can be transformed into each other by a finite sequence of diagonal flips. Recently Komuro gives a linear bound on the maximum number of diagonal flips needed for such a transformation. In this paper we show that any two labelled triangulations can be transformed into each other using at most O(nlogn) diagonal flips. We will also show that any planar triangulation with n>4 vertices has at least n− 2 flippable edges. Finally, we prove that if the minimum degree of a triangulation is at least 4, then it contains at least 2n + 3 flippable edges. These bounds can be achieved by an infinite class of triangulations. Received: June 3, 1998 Final version received: January 26, 2001     

A framework for advancing front techniques of finite element mesh generation

Advancing front techniques are a family of methods for finite element mesh generation that are particularly effective in dealing with complicated boundary geometries. In the first part of this paper, conditions are presented which ensure that any planar aft algorithm that meets these conditions terminates in a finite number of steps with a valid triangulation of the input domain. These conditions are described by specifying a framework of subtasks that can accommodate many aft methods and by prescribing the minimal requirements on each subtask that ensure correctness of an algorithm that conforms to the framework.An important efficiency factor in implementing an aft is the data structure used to represent the unmeshed regions during the execution of the algorithm. In the second part of the paper, we discuss the use of the constrained Delaunay triangulation as an efficient abstract data structure for the unmeshed regions. We indicate how the correctness conditions of the first part of the paper can be met using this representation. In this case, we also discuss the additional requirements on the framework which ensure that the generated mesh is a constrained Delaunay triangulation for the original boundary.The first author has been supported by CERFACS, Toulouse, France. Support was provided to the second author by the Natural Sciences and Engineering Research Council of Canada, and by the Information Technology Research Centre of Ontario.     

Bregman Voronoi Diagrams

The Voronoi diagram of a finite set of objects is a fundamental geometric structure that subdivides the embedding space into regions, each region consisting of the points that are closer to a given object than to the others. We may define various variants of Voronoi diagrams depending on the class of objects, the distance function and the embedding space. In this paper, we investigate a framework for defining and building Voronoi diagrams for a broad class of distance functions called Bregman divergences. Bregman divergences include not only the traditional (squared) Euclidean distance but also various divergence measures based on entropic functions. Accordingly, Bregman Voronoi diagrams allow one to define information-theoretic Voronoi diagrams in statistical parametric spaces based on the relative entropy of distributions. We define several types of Bregman diagrams, establish correspondences between those diagrams (using the Legendre transformation), and show how to compute them efficiently. We also introduce extensions of these diagrams, e.g., k-order and k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set of points and their connection with Bregman Voronoi diagrams. We show that these triangulations capture many of the properties of the celebrated Delaunay triangulation.     

Nitsche-mortaring for singularly perturbed convection–diffusion problems

In the present paper we analyse a finite element method for a singularly perturbed convection–diffusion problem with exponential boundary layers. Using a mortaring technique we combine an anisotropic triangulation of the layer region (into rectangles) with a shape regular one of the remainder of the domain. This results in a possibly non-matching (and hybrid), but layer adapted mesh of Shishkin type. We study the error of the method allowing different asymptotic behaviour of the triangulations and prove uniform convergence and a supercloseness property of the method. Numerical results supporting our analysis are presented.     

Delaunay triangulation and 3D adaptive mesh generation

Delaunay triangulation and its complementary structure the Voronoi polyhedra form two of the most fundamental constructs of computational geometry. Delaunay triangulation offers an efficient method for generating high-quality triangulations. However, the generation of Delaunay triangulations in 3D with Watson's algorithm, leads to the appearance of silver tetrahedra, in a relatively large percentage. A different method for generating high-quality tetrahedralizations, based on Delaunay triangulation and not presenting the problem of sliver tetrahedra, is presented. The method consists in a tetrahedra division procedure and an efficient method for optimizing tetrahedral meshes, based on the application of a set of topological Delaunay transformations for tetrahedra and a technique for node repositioning. The method is robust and can be applied to arbitrary unstructured tetrahedral meshes, having as a result the generation of high-quality adaptive meshes with varying density, totally eliminating the appearance of sliver elements. In this way it offers a convenient and highly flexible algorithm for implementation in a general purpose 3D adaptive finite element analysis system. Applications to various engineering problems are presented     

Degree-regular triangulations of torus and Klein bottle

A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In [8], Lutz has classified all the weakly regular triangulations on at most 15 vertices. In [5], Datta and Nilakantan have classified all the degree-regular triangulations of closed surfaces on at most 11 vertices. In this article, we have proved that any degree-regular triangulation of the torus is weakly regular. We have shown that there exists ann-vertex degree-regular triangulation of the Klein bottle if and only if n is a composite number ≥ 9. We have constructed two distinctn-vertex weakly regular triangulations of the torus for eachn ≥ 12 and a (4m + 2)-vertex weakly regular triangulation of the Klein bottle for eachm ≥ 2. For 12 ≤n ≤ 15, we have classified all then-vertex degree-regular triangulations of the torus and the Klein bottle. There are exactly 19 such triangulations, 12 of which are triangulations of the torus and remaining 7 are triangulations of the Klein bottle. Among the last 7, only one is weakly regular.     

An improvement of fixed point algorithms by using a good triangulation

We consider measures for triangulations ofR ⁿ. A new measure is introduced based on the ratio of the length of the sides and the content of the subsimplices of the triangulation. In a subclass of triangulations, which is appropriate for computing fixed points using simplicial subdivisions, the optimal one according to this measure is calculated and some of its properties are given. It is proved that for the average directional density this triangulation is optimal (within the subclass) asn goes to infinity. Furthermore, we compare the measures of the optimal triangulation with those of other triangulations. We also propose a new triangulation of the affine hull of the unit simplex. Finally, we report some computational experience that confirms the theoretical results.     

Sharp Concentration of the Number of Submaps in Random Planar Triangulations

We show that the maximum vertex degree in a random 3-connected planar triangulation is concentrated in an interval of almost constant width. This is a slightly weaker type of result than our earlier determination of the limiting distribution of the maximum vertex degree in random planar maps and in random triangulations of a (convex) polygon. We also derive sharp concentration results on the number of vertices of given degree in random planar maps of all three types. Some sharp concentration results about general submaps in 3-connected triangulations are also given.* Research supported by NSERC and Australian Research Council Research supported by the Australian Research Council     

On the number of Hamiltonian cycles in triangulations

It is proved that if a planar triangulation different from K₃ and K₄ contains a Hamiltonian cycle, then it contains at least four of them. Together with the result of Hakimi, Schmeichel, and Thomassen [2], this yields that, for n ? 12, the minimum number of Hamiltonian cycles in a Hamiltonian planar triangulation on n vertices is four. We also show that this theorem holds for triangulations of arbitrary surfaces and for 3-connected triangulated graphs.     

Nonconforming P1 elements on distorted triangulations: Lower bounds for the discrete energy norm error

Compared to conforming P1 finite elements, nonconforming P1 finite element discretizations are thought to be less sensitive to the appearance of distorted triangulations. E.g., optimal-order discrete H¹ norm best approximation error estimates for H² functions hold for arbitrary triangulations. However, the constants in similar estimates for the error of the Galerkin projection for second-order elliptic problems show a dependence on the maximum angle of all triangles in the triangulation. We demonstrate on an example of a special family of distorted triangulations that this dependence is essential, and due to the deterioration of the consistency error. We also provide examples of sequences of triangulations such that the nonconforming P1 Galerkin projections for a Poisson problem with polynomial solution do not converge or converge at arbitrarily low speed. The results complement analogous findings for conforming P1 finite elements.     

The hierarchical preconditioning on unstructured three-dimensional grids with locally refined regions

This paper presents two hierarchically preconditioned methods for the fast solution of mesh equations that approximate three-dimensional-elliptic boundary value problems on quasiuniform triangulations above all aiming at the numerical investigation of the previously suggested algorithms. Furthermore, improving the practical applicability of the methods unstructured three-dimensional grids possessing locally refined regions are considered. Based on the fictitious space approach, the original problem can be adaptively embedded into an auxiliary one in which hanging nodes occur. We implemented the corresponding Yserentant preconditioned conjugate gradient method as well as the BPX-preconditioned cg-iteration having nearly optimal computational costs. Several numerical examples demonstrate the efficiency of the artificially constructed hierarchical methods.     

Minimal triangulations of graphs: A survey

Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the result of adding an inclusion minimal set of edges to produce a triangulation. This topic was first studied from the standpoint of sparse matrices and vertex elimination in graphs. Today we know that minimal triangulations are closely related to minimal separators of the input graph. Since the first papers presenting minimal triangulation algorithms appeared in 1976, several characterizations of minimal triangulations have been proved, and a variety of algorithms exist for computing minimal triangulations of both general and restricted graph classes. This survey presents and ties together these results in a unified modern notation, keeping an emphasis on the algorithms.     

Computational Complexity of the Vertex Cover Problem in the Class of Planar Triangulations

We study the computational complexity of the vertex cover problem in the class of planar graphs (planar triangulations) admitting a plane representation whose faces are triangles. It is shown that the problem is strongly NP-hard in the class of 4-connected planar triangulations in which the degrees of vertices are of order O(log n), where n is the number of vertices, and in the class of plane 4-connected Delaunay triangulations based on the Minkowski triangular distance. A pair of vertices in such a triangulation is adjacent if and only if there is an equilateral triangle ?(p, λ) with p ∈ R² and λ > 0 whose interior does not contain triangulation vertices and whose boundary contains this pair of vertices and only it, where ?(p, λ) = p + λ? = {x ∈ R²: x = p + λa, a ∈ ?}; here ? is the equilateral triangle with unit sides such that its barycenter is the origin and one of the vertices belongs to the negative y-axis. Keywords: computational complexity, Delaunay triangulation, Delaunay TD-triangulation.     

On well-covered triangulations: Part II

A graph G is said to be well-covered if every maximal independent set of vertices has the same cardinality. A planar (simple) graph in which each face is a triangle is called a triangulation. It was proved in an earlier paper [A. Finbow, B. Hartnell, R. Nowakowski, M. Plummer, On well-covered triangulations: Part I, Discrete Appl. Math., 132, 2004, 97-108] that there are no 5-connected planar well-covered triangulations. It is the aim of the present paper to completely determine the 4-connected well-covered triangulations containing two adjacent vertices of degree 4. In a subsequent paper [A. Finbow, B. Hartnell, R. Nowakowski, M. Plummer, On well-covered triangulations: Part III (submitted for publication)], we show that every 4-connected well-covered triangulation contains two adjacent vertices of degree 4 and hence complete the task of characterizing all 4-connected well-covered planar triangulations. There turn out to be only four such graphs. This stands in stark contrast to the fact that there are infinitely many 3-connected well-covered planar triangulations.     

Asymptotic normality determined by high moments,and submap counts of random maps

We give a general result showing that the asymptotic behaviour of high moments determines the shape of distributions which are asymptotically normal. Both the factorial and non-factorial (non-central) moments are treated. This differs from the usual moment method in combinatorics, as the expected value may tend to infinity quite rapidly. Applications are given to submap counts in random planar triangulations, where we use a simple argument to asymptotically determine high moments for the number of copies of a given subtriangulation in a random 3-connected planar triangulation. Similar results are also obtained for 2-connected triangulations and quadrangulations with no multiple edges.Revised version: 6 February 2004Research supported by NSERCC and University of Macau.Research supported by the Australian Research Council and the Canada Research Chairs program. Research carried mainly while the author was at the Department of Mathematics and Statistics, University of Melbourne.