Optimality of the Delaunay triangulation in ℝ d

In this paper we present new optimality results for the Delaunay triangulation of a set of points in ℝ ^d . These new results are true in all dimensionsd. In particular, we define a power function for a triangulation and show that the Delaunay triangulation minimizes the power function over all triangulations of a point set. We use this result to show that (a) the maximum min-containment radius (the radius of the smallest sphere containing the simplex) of the Delaunay triangulation of a point set in ℝ ^d is less than or equal to the maximum min-containment radius of any other triangulation of the point set, (b) the union of circumballs of triangles incident on an interior point in the Delaunay triangulation of a point set lies inside the union of the circumballs of triangles incident on the same point in any other triangulation of the point set, and (c) the weighted sum of squares of the edge lengths is the smallest for Delaunay triangulation, where the weight is the sum of volumes of the triangles incident on the edge. In addition we show that if a triangulation consists of only self-centered triangles (a simplex whose circumcenter falls inside the simplex), then it is the Delaunay triangulation.     

A Monotonicity Property for Weighted Delaunay Triangulations

Given a triangulation of points in the plane and a function on the points, one may consider the Dirichlet energy, which is related to the Dirichlet energy of a smooth function. In fact, the Dirichlet energy can be derived from a finite element approximation. S. Rippa showed that the Dirichlet energy (which he refers to as the “roughness”) is minimized by the Delaunay triangulation by showing that each edge flip which makes an edge Delaunay decreases the energy. In this paper, we introduce a Dirichlet energy on a weighted triangulation which is a generalization of the energy on unweighted triangulations and an analogue of the smooth Dirichlet energy on a domain. We show that this Dirichlet energy has the property that each edge flip which makes an edge weighted Delaunay decreases the energy. The proof is done by a direct calculation, and so gives an alternate proof of Rippa’s result.     

Two characterisations of the minimal triangulations of permutation graphs

A minimal triangulation of a graph is a chordal supergraph with an inclusion-minimal edge set. Minimal triangulations are obtained from adding edges only to minimal separators, completing minimal separators into cliques. Permutation graphs are the comparability graphs whose complements are also comparability graphs. Permutation graphs can be characterised as the intersection graphs of specially arranged line segments in the plane, which is called a permutation diagram. The minimal triangulations of permutation graphs are known to be interval graphs, and they can be obtained from permutation diagrams by applying a geometric operation, that corresponds to the completion of separators into cliques. We precisely specify this geometric completion process to obtain minimal triangulations, and we completely characterise those interval graphs that are minimal triangulations of permutation graphs.     

Lattice triangulations of $\mathbb{E}^3 $ and of the 3-torus

This paper gives answers to a few questions concerning tilings of Euclidean spaces where the tiles are topological simplices with curvilinear edges. We investigate lattice triangulations of Euclidean 3-space in the sense that the vertices form a lattice of rank 3 and such that the triangulation is invariant under all translations of that lattice. This is the dual concept of a primitive lattice tiling where the tiles are not assumed to be Euclidean polyhedra but only topological polyhedra. In 3-space there is a unique standard lattice triangulation by Euclidean tetrahedra (and with straight edges) but there are infinitely many non-standard lattice triangulations where the tetrahedra necessarily have certain curvilinear edges. From the view-point of Discrete Differential Geometry this tells us that there are such triangulations of 3-space which do not carry any flat discrete metric which is equivariant under the lattice. Furthermore, we investigate lattice triangulations of the 3-dimensional torus as quotients by a sublattice. The standard triangulation admits such quotients with any number n ≥ 15 of vertices. The unique one with 15 vertices is neighborly, i.e., any two vertices are joined by an edge. It turns out that for any odd n ≥ 17 there is an n-vertex neighborly triangulation of the 3-torus as a quotient of a certain non-standard lattice triangulation. Combinatorially, one can obtain these neighborly 3-tori as slight modifications of the boundary complexes of the cyclic 4-polytopes. As a kind of combinatorial surgery, this is an interesting construction by itself.     

Enumerating edge-constrained triangulations and edge-constrained non-crossing geometric spanning trees

In this paper, we present algorithms for enumerating, without repetitions, all triangulations and non-crossing geometric spanning trees on a given set of n points in the plane under edge inclusion constraint (i.e., some edges are required to be included in the graph). We will first extend the lexicographically ordered triangulations introduced by Bespamyatnikh to the edge-constrained case, and then we prove that a set of all edge-constrained non-crossing spanning trees is connected via remove-add flips, based on the edge-constrained lexicographically largest triangulation. More specifically, we prove that all edge-constrained triangulations can be transformed to the lexicographically largest triangulation among them by O(n²) greedy flips, i.e., by greedily increasing the lexicographical ordering of the edge list, and a similar result also holds for a set of edge-constrained non-crossing spanning trees. Our enumeration algorithms generate each output triangulation and non-crossing spanning tree in O(loglogn) and O(n²) time, respectively, based on the reverse search technique.     

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.     

Generating realistic terrains with higher-order Delaunay triangulations

For hydrologic applications, terrain models should have few local minima, and drainage lines should coincide with edges. We show that triangulating a set of points with elevations such that the number of local minima of the resulting terrain is minimized is NP-hard for degenerate point sets. The same result applies when there are no degeneracies for higher-order Delaunay triangulations. Two heuristics are presented to reduce the number of local minima for higher-order Delaunay triangulations, which start out with the Delaunay triangulation. We give efficient algorithms for their implementation, and test on real-world data how well they perform. We also study another desirable drainage characteristic, few valley components, and how to obtain it for higher-order Delaunay triangulations. This gives rise to a third heuristic. Tables and visualizations show how the heuristics perform for the drainage characteristics on real-world data.     

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.     

Approximating Minimum-Weight Triangulations in Three Dimensions

Let S be a set of noncrossing triangular obstacles in R ³ with convex hull H . A triangulation T of H is compatible with S if every triangle of S is the union of a subset of the faces of T. The weight of T is the sum of the areas of the triangles of T. We give a polynomial-time algorithm that computes a triangulation compatible with S whose weight is at most a constant times the weight of any compatible triangulation. One motivation for studying minimum-weight triangulations is a connection with ray shooting. A particularly simple way to answer a ray-shooting query (``Report the first obstacle hit by a query ray') is to walk through a triangulation along the ray, stopping at the first obstacle. Under a reasonably natural distribution of query rays, the average cost of a ray-shooting query is proportional to triangulation weight. A similar connection exists for line-stabbing queries (``Report all obstacles hit by a query line'). Received February 3, 1997, and in revised form August 21, 1998.     

An efficient algorithm for enumeration of triangulations

We consider the problem of enumerating triangulations of n points in the plane in general position. We introduce a tree of triangulations and present an algorithm for enumerating triangulations in O(loglogn) time per triangulation. It improves the previous bound by almost linear factor.     

Lepp-bisection algorithms,applications and mathematical properties

Longest edge (nested) algorithms for triangulation refinement in two dimensions are able to produce hierarchies of quality and nested irregular triangulations as needed both for adaptive finite element methods and for multigrid methods for PDEs. In addition, right-triangle bintree triangulations are multiresolution algorithms used for terrain modeling and real time visualization of terrain applications. These algorithms are based on the properties of the consecutive bisection of a triangle by the median of the longest edge, and can be formulated in terms of the longest edge propagation path (Lepp) and terminal edge concepts, which implies the use of very local refinement operations over fully conforming meshes (where the intersection of pairs of neighbor triangles is either a common edge or a common vertex). In this paper we review the Lepp-bisection algorithms, their properties and applications. To the end we use recent simpler and stronger results on the complexity aspects of the bisection method and its geometrical properties. We discuss and analyze the computational costs of the algorithms. The generalization of the algorithms to 3-dimensions is also discussed. Applications of these methods are presented: for serial and parallel view dependent level of detail terrain rendering, and for the parallel refinement of tetrahedral meshes.     

Delaunay Triangulation of Manifolds

We present an algorithm for producing Delaunay triangulations of manifolds. The algorithm can accommodate abstract manifolds that are not presented as submanifolds of Euclidean space. Given a set of sample points and an atlas on a compact manifold, a manifold Delaunay complex is produced for a perturbed point set provided the transition functions are bi-Lipschitz with a constant close to 1, and the original sample points meet a local density requirement; no smoothness assumptions are required. If the transition functions are smooth, the output is a triangulation of the manifold. The output complex is naturally endowed with a piecewise-flat metric which, when the original manifold is Riemannian, is a close approximation of the original Riemannian metric. In this case the output complex is also a Delaunay triangulation of its vertices with respect to this piecewise-flat metric.     

Unfolding of Surfaces

This paper deals with the approximation of the unfolding of a smooth globally developable surface (i.e. "isometric" to a domain of ) with a triangulation. We prove the following result: let T_n be a sequence of globally developable triangulations which tends to a globally developable smooth surface S in the Hausdorff sense. If the normals of T_n tend to the normals of S, then the shape of the unfolding of T_n tends to the shape of the unfolding of S. We also provide several examples: first, we show globally developable triangulations whose vertices are close to globally developable smooth surfaces; we also build sequences of globally developable triangulations inscribed on a sphere, with a number of vertices and edges tending to infinity. Finally, we also give an example of a triangulation with strictly negative Gauss curvature at any interior point, inscribed in a smooth surface with a strictly positive Gauss curvature. The Gauss curvature of these triangulations becomes positive (at each interior vertex) only by switching some of their edges.     

Multithread parallelization of Lepp-bisection algorithms

Longest edge (nested) algorithms for triangulation refinement in two dimensions are able to produce hierarchies of quality and nested irregular triangulations as needed both for adaptive finite element methods and for multigrid methods. They can be formulated in terms of the longest edge propagation path (Lepp) and terminal edge concepts, to refine the target triangles and some related neighbors. We discuss a parallel multithread algorithm, where every thread is in charge of refining a triangle t and its associated Lepp neighbors. The thread manages a changing Lepp(t) (ordered set of increasing triangles) both to find a last longest (terminal) edge and to refine the pair of triangles sharing this edge. The process is repeated until triangle t is destroyed. We discuss the algorithm, related synchronization issues, and the properties inherited from the serial algorithm. We present an empirical study that shows that a reasonably efficient parallel method with good scalability was obtained.     

Flipping Edges in Triangulations

In this paper we study the problem of flipping edges in triangulations of polygons and point sets. One of the main results is that any triangulation of a set of n points in general position contains at least edges that can be flipped. We also prove that O(n + k ² ) flips are sufficient to transform any triangulation of an n -gon with k reflex vertices into any other triangulation. We produce examples of n -gons with triangulations T and T' such that to transform T into T' requires Ω(n ² ) flips. Finally we show that if a set of n points has k convex layers, then any triangulation of the point set can be transformed into any other triangulation using at most O(kn) flips. Received May 13, 1997, and in revised form July 21, 1998, and February 1, 1999.     

Irreducible Triangulations of Surfaces with Boundary

A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g ≥ 0 with b ≥ 0 boundary components is O(g + b). So far, the result was known only for surfaces without boundary (b = 0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary.     

Approximating the minimum weight steiner triangulation

We show that the length of the minimum weight Steiner triangulation (MWST) of a point set can be approximated within a constant factor by a triangulation algorithm based on quadtrees. InO(n logn) time we can compute a triangulation withO(n) new points, and no obtuse triangles, that approximates the MWST. We can also approximate the MWST with triangulations having no sharp angles. We generalize some of our results to higher-dimensional triangulation problems. No previous polynomial-time triangulation algorithm was known to approximate the MWST within a factor better thanO(logn).     

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 $\Delta$.