A new asymmetric inclusion region for minimum weight triangulation

As a global optimization problem, planar minimum weight triangulation problem has attracted extensive research attention. In this paper, a new asymmetric graph called one-sided β-skeleton is introduced. We show that the one-sided circle-disconnected (?2b){(\sqrt{2}\beta)} -skeleton is a subgraph of a minimum weight triangulation. An algorithm for identifying subgraph of minimum weight triangulation using the one-sided (?2b){(\sqrt{2}\beta)} -skeleton is proposed and it runs in O(n^4/3+e+min{klogn, n²logn}){O(n^{4/3+\epsilon}+\min\{\kappa \log n, n^2\log n\})} time, where κ is the number of intersected segmented between the complete graph and the greedy triangulation of the point set.     

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.     

Improved heuristics for the minimum weight triangulation problem

Given a planar point setS, a triangulation ofS is a maximal set of non-intersecting line segments connecting the points. The minimum weight triangulation problem is to find a triangulation ofS such that the sum of the lengths of the line segments in it is the smallest. No polynomial time algorithm is known to produce the optimal or even a constant approximation of the optimal solution, and it is also unknown whether the problem is NP-hard. In this paper, we propose two improved heuristics, which triangulate a set ofn points in a plane inO(n ³) time and never do worse than the minimum spanning tree triangulation algorithm given by Lingas and the greedy spanning tree triangulation algorithm given by Heath and Pemmaraju. These two algorithms both produce an optimal triangulation if the points are the vertices of a convex polygon, and also do the same in some special cases.     

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).     

Minimal triangulation of a graph and optimal pivoting order in a sparse matrix

This paper considers the problem of finding a minimal triangulation of an undirected graph G = (V, E), where a triangulation is a set T such that every cycle in G = (V, E ∪ T) has a chord. A triangulation T is minimal (minimum) if no triangulation F exists such that F is a proper subset of $\text{T (¦F¦ < ¦T¦)}$, and an ordering α is optimal (optimum) if a minimal (minimum) triangulation is generated by α. A minimum triangulation (optimum ordering) is necessarily minimal (optimal), but the converse is not necessarily true. A necessary and sufficient condition for a triangulation to be minimal is presented. This leads to an algorithm for finding an optimal ordering α which produces a minimal set of “fill-in” when the process is viewed as triangular factorization of a sparse matrix.     

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.     

A new subdivision for computing fixed points with a homotopy algorithm

In this paper a triangulation is introduced for homotopy methods to compute fixed points on the unit simplex or inR ⁿ . This triangulation allows for factors of incrementation of more than two. The factor may be of any size and even different at each level. Also the starting point on a new level may be any gridpoint of the last found completely labelled subsimplex on the last level. So, the decision which new factor of incrementation and which starting point is used, can be made on the ground of previous approximations. Doing so, the convergence rate can be accelerated without using restart methods.     

关于平面点集三角剖分的两类子集

给定欧氏平面上的一个点集合Ｓ，我们给出两类端点在Ｓ中的线段集合，第一类线段集合是Ｓ的任一三角剖分的子集，第二类线段集合是Ｓ的任一最小权三解剖分的子集，这两类子集是不相交的，这两类子集合的计算要用Ｏ(n³)时间和Ｏ（ｎ）空间．     

The smallest point of a polytope

This note suggests new ways for calculating the point of smallest Euclidean norm in the convex hull of a given set of points inR ⁿ . It is shown that the problem can be formulated as a linear least-square problem with nonnegative variables or as a least-distance problem. Numerical experiments illustrate that the least-square problem is solved efficiently by the active set method. The advantage of the new approach lies in the solution of large sparse problems. In this case, the new formulation permits the use of row relaxation methods. In particular, the least-distance problem can be solved by Hildreth's method.     

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.     

Quasi-Greedy Triangulations Approximating the Minimum Weight Triangulation

This article settles the following two longstanding open problems:

• What is the worst case approximation ratio between the greedy triangulation and the minimum weight triangulation?

• Is there a polynomial time algorithm that always produces a triangulation whose length is within a constant factor from the minimum?

The answer to the first question is that the known lower bound is tight. The second question is answered in the affirmative by using a slight modification of anO(n log n) algorithm for the greedy triangulation. We also derive some other interesting results. For example, we show that a constant-factor approximation of the minimum weight convex partition can be obtained within the same time bounds.     

A Large Subgraph of the Minimum Weight Triangulation

We present an O(n ⁴ )-time and O(n ² )-space algorithm that computes a subgraph of the minimum weight triangulation (MWT) of a general point set. The algorithm works by finding a collection of edges guaranteed to be in any locally minimal triangulation. We call this subgraph the LMT-skeleton. We also give a variant called the modified LMT-skeleton that is both a more complete subgraph of the MWT and is faster to compute requiring only O(n ² ) time and O(n) space in the expected case for uniform distributions. Several experimental implementations of both approaches have shown that for moderate-sized point sets (up to 350 points^1) the skeletons are connected, enabling an efficient completion of the exact MWT. We are thus able to compute the MWT of substantially larger random point sets than have previously been computed. ^1Though in this paper we summarize some empirical findings for input sets of up to 350 points, a variant of the algorithm has been implemented and tested on up to 40,000 points producing connected subgraphs [2]. Received May 29, 1996, and in revised form January 10, 1997.     

Manifold Reconstruction in Arbitrary Dimensions Using Witness Complexes

It is a well-established fact that the witness complex is closely related to the restricted Delaunay triangulation in low dimensions. Specifically, it has been proved that the witness complex coincides with the restricted Delaunay triangulation on curves, and is still a subset of it on surfaces, under mild sampling conditions. In this paper, we prove that these results do not extend to higher-dimensional manifolds, even under strong sampling conditions such as uniform point density. On the positive side, we show how the sets of witnesses and landmarks can be enriched, so that the nice relations that exist between restricted Delaunay triangulation and witness complex hold on higher-dimensional manifolds as well. We derive from our structural results an algorithm that reconstructs manifolds of any arbitrary dimension or co-dimension at different scales. The algorithm combines a farthest-point refinement scheme with a vertex pumping strategy. It is very simple conceptually, and it does not require the input point sample to be sparse. Its running time is bounded by c(d)n ², where n is the size of the input point cloud, and c(d) is a constant depending solely (yet exponentially) on the dimension d of the ambient space. Although this running time makes our reconstruction algorithm rather theoretical, recent work has shown that a variant of our approach can be made tractable in arbitrary dimensions, by building upon the results of this paper. This work was done while S.Y. Oudot was a post-doctoral fellow at Stanford University. His email there is no longer valid.     

A minimal triangulation of complex projective plane admitting a chess colouring of four-dimensional simplices

We construct and study a new 15-vertex triangulation X of the complex projective plane ℂP². The automorphism group of X is isomorphic to S ₄ × S ₃. We prove that the triangulation X is the minimal (with respect to the number of vertices) triangulation of ℂP² admitting a chess colouring of four-dimensional simplices. We provide explicit parametrizations for the simplices of X and show that the automorphism group of X can be realized as a group of isometries of the Fubini-Study metric. We find a 33-vertex subdivision $ \bar X $ \bar X of the triangulation X such that the classical moment mapping μ: ℂP² → Δ² is a simplicial mapping of the triangulation $ \bar X $ \bar X onto the barycentric subdivision of the triangle Δ². We study the relationship of the triangulation X with complex crystallographic groups.     

Approximation for Minimum Triangulations of Simplicial Convex 3-Polytopes

A minimum triangulation of a convex 3-polytope is a triangulation that contains the minimum number of tetrahedra over all its possible triangulations. Since finding minimum triangulations of convex 3-polytopes was recently shown to be NP-hard, it becomes significant to find algorithms that give good approximation. In this paper we give a new triangulation algorithm with an improved approximation ratio 2 - Ω(1/\sqrt n ) , where n is the number of vertices of the polytope. We further show that this is the best possible for algorithms that only consider the combinatorial structure of the polytopes. Received August 5, 2000, and in revised form March 29, 2001, and May 3, 2001. Online publication October 12, 2001.     

Generalized delaunay triangulation for planar graphs

We introduce the notion of generalized Delaunay triangulation of a planar straight-line graphG=(V, E) in the Euclidean plane and present some characterizations of the triangulation. It is shown that the generalized Delaunay triangulation has the property that the minimum angle of the triangles in the triangulation is maximum among all possible triangulations of the graph. A general algorithm that runs inO(|V|²) time for computing the generalized Delaunay triangulation is presented. When the underlying graph is a simple polygon, a divide-and-conquer algorithm based on the polygon cutting theorem of Chazelle is given that runs inO(|V| log |V|) time.Supported in part by the National Science Foundation under Grants DCR 8420814 and ECS 8340031.     

\tilde K'_3 —A new triangulation ofR n—A new triangulation ofR n

This paper introduces a new triangulation ofR ⁿ. The triangulation is to be applied for solving system of nonlinear equation through a fixed point technique.     

A class of simplicial restart fixed point algorithms without an extra dimension

In an earlier paper we introduced an algorithm for approximating a fixed point of a mapping on the product space of unit simplices. Ideas of that paper are used to construct a class of triangulations ofR ⁿ. More precisely, for somek, 1 k n, and positive integersm ₁ , m_k with sumn, a triangulation ofR ⁿ is obtained by triangulating the cells which are formed by taking the product of given triangulations ofR ^mj, j = 1, ,k. The triangulation of each cell will be defined in relation to an arbitrarily chosen pointv inR ⁿ, being the starting point of the algorithm. Fork = n we obtain theK triangulation originally due to Todd. Each element of the class can be used to find a simplex which approximates a fixed point of a mapping onR ⁿ by generating a unique path of adjacent simplices of variable dimension starting with the pointv. We also give convergence conditions. It is indicated how in casek = n a connected set of fixed points can be generated. Moreover, we give some computational experience.     

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.     

Solving discrete systems of nonlinear equations

We study the existence problem of a zero point of a function defined on a finite set of elements of the integer lattice Zⁿ of the n-dimensional Euclidean space Rⁿ. It is assumed that the set is integrally convex, which implies that the convex hull of the set can be subdivided in simplices such that every vertex is an element of Zⁿ and each simplex of the triangulation lies in an n-dimensional cube of size one. With respect to this triangulation we assume that the function satisfies some property that replaces continuity. Under this property and some boundary condition the function has a zero point. To prove this we use a simplicial algorithm that terminates with a zero point within a finite number of iterations. The standard technique of applying a fixed point theorem to a piecewise linear approximation cannot be applied, because the ‘continuity property’ is too weak to assure that a zero point of the piecewise linear approximation induces a zero point of the function itself. We apply the main existence result to prove the existence of a pure Cournot-Nash equilibrium in a Cournot oligopoly model. We further obtain a discrete analogue of the well-known Borsuk-Ulam theorem and a theorem for the existence of a solution for the discrete nonlinear complementarity problem.