Delaunay Graph Research Articles

A Validity Index for Prototype-Based Clustering of Data Sets With Complex Cluster Structures

Evaluation of how well the extracted clusters fit the true partitions of a data set is one of the fundamental challenges in unsupervised clustering because the data structure and the number of clusters are unknown a priori. Cluster validity indices are commonly used to select the best partitioning from different clustering results; however, they are often inadequate unless clusters are well separated or have parametrical shapes. Prototype-based clustering (finding of clusters by grouping the prototypes obtained by vector quantization of the data), which is becoming increasingly important for its effectiveness in the analysis of large high-dimensional data sets, adds another dimension to this challenge. For validity assessment of prototype-based clusterings, previously proposed indexes-mostly devised for the evaluation of point-based clusterings-usually perform poorly. The poor performance is made worse when the validity indexes are applied to large data sets with complicated cluster structure. In this paper, we propose a new index, Conn_Index, which can be applied to data sets with a wide variety of clusters of different shapes, sizes, densities, or overlaps. We construct Conn_Index based on inter- and intra-cluster connectivities of prototypes. Connectivities are defined through a "connectivity matrix", which is a weighted Delaunay graph where the weights indicate the local data distribution. Experiments on synthetic and real data indicate that Conn_Index outperforms existing validity indices, used in this paper, for the evaluation of prototype-based clustering results.

Witness (Delaunay) graphs

Proximity graphs are used in several areas in which a neighborliness relationship for input data sets is a useful tool in their analysis, and have also received substantial attention from the graph drawing community, as they are a natural way of implicitly representing graphs. However, as a tool for graph representation, proximity graphs have some limitations that may be overcome with suitable generalizations. We introduce a generalization, witness graphs, that encompasses both the goal of more power and flexibility for graph drawing issues and a wider spectrum for neighborhood analysis. We study in detail two concrete examples, both related to Delaunay graphs, and consider as well some problems on stabbing geometric objects and point set discrimination, that can be naturally described in terms of witness graphs.

A Method of Using Sub Graph Based on the Characteristics of the Optima for the TSPs

Using the characteristics of the optima is so effective to solve the traveling salesman problems. In our study, the edges on the optimal tours calculated completely have been analyzed to have the characteristics of their mutual relationship. The edge set based on the characteristics can construct the sub graph including all or almost the edges selected as optimal edges. If a sub graph has all or almost the edges of constructing the optimum, the investigation on this graph will lead to discover a good solution in reasonable time. Therefore, such sub graphs are the effective tools for existing methods to become improved methods. In this paper, Delaunay graph and two graphs, called Fragmented Nearest Neighbor graph, are proposed as the special sub graphs. And the experiments by the simulated annealing method that is executable on sub graph explain that Delaunay graph as a triangulated graph is the most effective graph among the three sub graphs to find optimum or sub optimum tour of the traveling salesman problems.

Variational surface reconstruction based on Delaunay triangulation and graph cut

AbstractA novel method for surface reconstruction from an unorganized point set is presented. An energy functional based on a weighted minimal surface model is proposed for surface reconstruction, which is efficiently minimized by graph cut methods. By solving the minimization problem on the graph dual to a Delaunay‐based tetrahedral mesh, the advantages of explicit and implicit methods for surface reconstruction are well integrated. A triangular surface mesh homeomorphic to the original surface can be extracted directly from the tetrahedral mesh provided a sufficient sampling density exists. Difficult cases involving undersampling, non‐uniformity, noises and topological complexities can be handled effectively as well. Furthermore, for the first time, multi‐phase surface reconstruction is realized based on the graph cut methods. Various examples are included for demonstrating the efficiency and effectiveness of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.

A study of gridless method with dynamic clouds of points for solving unsteady CFD problems in aerodynamics

AbstractWhen solving unsteady computational fluid dynamics problems in aerodynamics with a gridless method, a cloud of points is usually required to be regenerated due to its accommodation to moving boundaries. In order to handle this problem conveniently, a fast dynamic cloud method based on Delaunay graph mapping strategy is proposed in this paper. A dynamic cloud method makes use of algebraic mapping principles and therefore points can be accurately redistributed in the flow field without any iteration. In this way, the structure of the gridless clouds is not necessarily changed so that the clouds regeneration can be avoided successfully. The spatial derivatives of the mathematical modeling of the flow are directly determined by using weighted least‐squares method in each cloud of points, and then numerical fluxes can be obtained. A dual time‐stepping method is further implemented to advance the two‐dimensional Euler equations in arbitrary Lagarangian–Eulerian formulation in time. Finally, unsteady transonic flows over two different oscillating airfoils are simulated with the above method and results obtained are in good agreement with the experimental data. Copyright © 2009 John Wiley & Sons, Ltd.

On the stretch factor of convex Delaunay graphs

Let C be a compact and convex set in the plane that contains the origin in its interior, and let S be a finite set of points in the plane. The Delaunay graph DG C ( S ) of S is defined to be the dual of the Voronoi diagram of S with respect to the convex distance function defined by C . We prove that DG C ( S ) is a t -spanner for S , for some constant t that depends only on the shape of the set C . Thus, for any two points p and q in S , the graph DG C ( S ) contains a path between p and q whose Euclidean length is at most t times the Euclidean distance between p and q .

AEROELASTIC ANALYSIS OF ROTOR BLADES USING CFD/CSD COUPLING IN HOVER MODE

A computational fluid dynamics (CFD) is coupled with a computational structural dynamics (CSD) to simulate the unsteady rotor flow with aeroelasticity effects. An unstructured upwind Navier-Stokes solver was developed for this simulation, with 2nd order time-accurate dual-time stepping method for temporal discretization and low Mach number preconditioning method. For turbulent flows, both the Spalart-Allmaras and Menter's SST model are available. Mesh deformation is achieved through a fast dynamic grid method called Delaunay graph map method for unsteady flow simulation. The rotor blades are modeled as Hodges & Dowell's nonlinear beams coupled flap-lag-torsion. The rotorcraft computational structural dynamics code employs the 15-dof beam finite element formulation for modeling. The structure code was validated by comparing the natural frequencies of a rotor model with UMARC. The flow and structure codes are coupled tightly with information exchange several times at every time step. A rotor blade model's unsteady flow field in the hover mode is simulated using the coupling method. Effect of blade elasticity with aerodynamic loads was compared with rigid blade.

An Edge-Constrained Localized Delaunay Graph for Geographic Routing in Mobile Ad Hoc and Sensor Networks

In mobile ad hoc and sensor networks, since all nodes are mobile and there is no fixed infrastructure, the design of routing protocols becomes one of the most challenging issues. In recent years, geographic routing protocols have been widely used. Most of them, e.g., greedy-face-greedy routing protocols, need nodes to construct planar graphs as the underlying graphs. In this paper, we propose an edge constrained localized Delaunay graph, denoted by ECLDel, as the underlying graph for geographic routing. We prove that the ECLDel is a planar t-spanner of the unit-disk graph. Geographic routing on ECLDel is as efficient as on the previous work of PLDel in terms of path length (hop count). However, the construction of ECLDel graph is far more simple and it converges faster. This is due to the following two reasons: First, we significantly reduce the number of messages broadcast by each node from five rounds (each round may contain several messages) to only two messages; second, we define two new types of edges, the intersecting Gabriel (IG) edges and the unaware intersection (UI) edges, which are constrained in the ECLDel graph. These edges help significantly reduce the size of messages broadcast by each node. The decrease of both the number and the size of messages broadcast by each node reduces the communication cost, and saves the network bandwidth and node power, which are desirable for mobile ad hoc and sensor networks. Our simulation results show that the average number of messages and the average size of messages broadcast by each node are, respectively, 65 and 42 percent less in the construction of ECLDel than that in PLDel.

Coverage Optimization and Spatial Load Balancing by Robotic Sensor Networks

This technical note studies robotic sensor networks performing static coverage optimization with area constraints. Given a density function describing the probability of events happening and a performance function measuring the cost to service a location, the objective is to position sensors in the environment so as to minimize the expected servicing cost. Moreover, because of load balancing considerations, the area of the region assigned to each robot is constrained to be a pre-specified amount. We characterize the optimal configurations as center generalized Voronoi configurations. The generalized Voronoi partition depends on a set of weights, one per robot, assigned to the network. We design a Jacobi iterative algorithm to find the weight assignment whose corresponding generalized Voronoi partition satisfies the area constraints. This algorithm is distributed over the generalized Delaunay graph. We also design the ?move-to-center-and-compute-weight? strategy to steer the robotic network towards the set of center generalized Voronoi configurations while monotonically optimizing coverage.

Unicycle Coverage Control Via Hybrid Modeling

This technical note presents gradient-descent coverage algorithms for a group of nonholonomic vehicles. Similarly to previous approaches, the deployment strategy relies on Locational Optimization techniques and algorithms are distributed in the sense of the Delaunay graph. In order to deal with unicycle dynamics and guarantee performance, we introduce several vehicle modes and integrate them in a hybrid system. We then analyze the algorithms with a recently introduced invariance principle for hybrid systems.

RF-Power Overlapping Control for Connectivity Awareness in Wireless Ad-Hoc and Sensor Networks

In this paper a novel power control scheme for establishing end-to-end connectivity over Wireless Ad-Hoc and Sensor Networks (WASN) is presented. Based on a deterministic connectivity model, the proposed scheme exploits overlapping information related to the properties of the network's Delaunay graph and localization. Sufficient conditions for establishing end-to-end connectivity over the WASN are provided; these are strongly attached to the properties of the network's Delaunay graph, formed by the nodes’ spatial distribution. Under the provision of the Delaunay-related overlapping information to the network's operating nodes, these sufficient conditions are utilized for the construction of a distributed Shortest Path-like algorithm. Without any additional network interaction, the resulting Distributed Delaunay Connectivity Algorithm (DDelCA) is capable of adjusting the nodal transmission power to the minimum value that establishes the network's end-to-end connectivity. Extensive simulation results are offered to evaluate the network's performance using the DDelCA transmission power configuration and highlight the benefits of the DDelCA scheme.

ON STRUCTURAL AND GRAPH THEORETIC PROPERTIES OF HIGHER ORDER DELAUNAY GRAPHS

Given a set P of n points in the plane, the order-k Delaunay graph is a graph with vertex set P and an edge exists between two points p, q ∈ P when there is a circle through p and q with at most k other points of P in its interior. We provide upper and lower bounds on the number of edges in an order-k Delaunay graph. We study the combinatorial structure of the set of triangulations that can be constructed with edges of this graph. Furthermore, we show that the order-k Delaunay graph is connected under the flip operation when k ≤ 1 but not necessarily connected for other values of k. If P is in convex position then the order-k Delaunay graph is connected for all k ≥ 0. We show that the order-k Gabriel graph, a subgraph of the order-k Delaunay graph, is Hamiltonian for k ≥ 15. Finally, the order-k Delaunay graph can be used to efficiently solve a coloring problem with applications to frequency assignments in cellular networks.

Analysis of the Spatial Distribution of Galaxies in the Numerical Galaxy Catalog

Abstract We examine the spatial distribution of galaxies in the Numerical Galaxy Catalog, which is based on a hierarchical clustering framework. This catalog is constructed from a semianalytic model of galaxy formation combined with high-resolution $N$-body simulations in a $\Lambda$-dominated flat cold dark-matter cosmological model. To quantify the spatial distribution of galaxies, we use a graph-theoretical approach, because it is one of the most powerful statistical ways of estimating spatial data. In particular, three types of graphs are adopted: the Delaunay graph, the minimal spanning tree, and the constellation graph. To quantify the galaxy distributions, we apply statistical measures suitable for each type. The mass distributions in the cold dark-matter universe are also examined to clarify differences in the spatial distribution of dark matter and galaxies. The spatial distribution of galaxies in the two-degree field (2dF) galaxy redshift survey is finally compared with that in the Numerical Galaxy Catalog. From our analysis, we definitely show that galaxy distributions in the Numerical Galaxy Catalog are different from the dark-matter distributions. We also find that the Numerical Galaxy Catalog considerably improves the theoretical prediction of spatial galaxy distributions, although it does not do enough to reproduce the 2dF galaxy redshift survey. Finally, we show that the constellation graph and the minimal spanning tree are convenient for quantifing the galaxy distributions in an objective manner.

COMBINATION OF MESH DEFORMATION AND OVERSET GRID FOR SIMULATING UNSTEADY FLOW OF INSECT FLIGHT

As numerical simulation of unsteady flows due to moving boundaries such as flexible flapping-wings is difficult by conventional approaches, an effective strategy which combines mesh deformation based on Delaunay graph mapping and unstructured overset grids is proposed in this paper. A Delaunay graph is generated for each body-fitted grid cluster which overlaps or is embedded within an off-body background grid cluster. At each time step, the graph moves according to the wing's motion and deformation, and the grids move to new positions according to a one-to-one mapping between the graph and the grid. Then, intergrid-boundary definition is implemented automatically for computation.

THE PREDICATES FOR THE EXACT VORONOI DIAGRAM OF ELLIPSES UNDER THE EUCLIDIEAN METRIC

This article examines the computation of the Delaunay graph and its dual Voronoi diagram of a set of ellipses in the Euclidean plane. We propose the first complete methods, under the exact computation paradigm, for the predicates of an incremental algorithm: κ1 decides which one of two given ellipses is closest to a given exterior point; κ2 decides the position of a query ellipse relative to an external bitangent line of two given ellipses; κ3 decides the position of a query ellipse relative to a Voronoi circle of three given ellipses; κ4 determines the type of conflict between a Voronoi edge, defined by four given ellipses, and a query ellipse. The article is restricted to non-intersecting ellipses, but the extension to arbitrary ones is possible. The ellipses are input in parametric representation, i.e., constructively in terms of their axes, center and rotation. For κ1 and κ2 we derive algebraic conditions optimal in terms of the degree of the algebraic numbers involved, and provide efficient implementations in C++ . For κ3 we compute a tight bound on the number of complex tritangent circles and design an exact symbolic-numeric algorithm, which is implemented in MAPLE. This essentially answers κ4 as well. We conclude with further work on lifting the condition of non-intersecting ellipses.

Rule-based programming for integrative biological modeling

Systems biology aims at integrating processes at various time and spatial scales into a single and coherent formal description to allow computer modeling. In this context, we focus on rule-based modeling and its integration in the domain-specific language MGS . Through the notions of topological collections and transformations, MGS allows the modeling of biological processes at various levels of description. We validate our approach through the description of various models of the genetic switch of the λ phage, from a very simple biochemical description of the process to an individual-based model on a Delaunay graph topology. This approach is a first step into providing the requirements for the emerging field of spatial systems biology which integrates spatial properties into systems biology.

Delaunay graphs of point sets in the plane with respect to axis‐parallel rectangles

AbstractGiven a point set P in the plane, the Delaunay graph with respect to axis‐parallel rectangles is a graph defined on the vertex set P, whose two points p,q ∈ P are connected by an edge if and only if there is a rectangle parallel to the coordinate axes that contains p and q, but no other elements of P. The following question of Even et al. (SIAM J Comput 33 (2003) 94–136) was motivated by a frequency assignment problem in cellular telephone networks: Does there exist a constant c > 0 such that the Delaunay graph of any set of n points in general position in the plane contains an independent set of size at least cn? We answer this question in the negative, by proving that the largest independent set in a randomly and uniformly selected point set in the unit square is O(nlog 2 log n/log n), with probability tending to 1. We also show that our bound is not far from optimal, as the Delaunay graph of a uniform random set of n points almost surely has an independent set of size at least cn log log n/(log n log log log n).We give two further applications of our methods: (1) We construct two‐dimensional n‐element partially ordered sets such that the size of the largest independent sets of vertices in their Hasse diagrams is o(n). This answers a question of Matoušek and Přívětivý (Combinat Probab Comput 15 (2006) 473–475) and improves a result of Kříž and Nešetřil (Order 8 (1991) 41–48). (2) For any positive integers c and d, we prove the existence of a planar point set with the property that no matter how we color its elements by c colors, we find an axis‐parallel rectangle containing at least d points, all of which have the same color. This solves an old problem from the work of Brass et al. (Research Problem in Discrete Geometry Springer‐Verlag, New York, 2005). © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009

R-Local Delaunay Inhibition Model

Unlike in the classical framework of Gibbs point processes (usually acting on the complete graph), in the context of nearest-neighbour Gibbs point processes the nonnegativeness of the interaction functions do not ensure the local stability property. This paper introduces domain-wise (but not pointwise) inhibition stationary Gibbs models based on some tailor-made Delaunay subgraphs. All of them are subgraphs of the R-local Delaunay graph, defined as the Delaunay subgraph specifically not containing the edges of Delaunay triangles with circumscribed circles of radii greater than some large positive real value R. The usual relative compactness criterion for point processes needed for the existence result is directly derived from the Ruelle-bound of the correlation functions. Furthermore, assuming only the nonnegativeness of the energy function, we have managed to prove the existence of the existence of R-local Delaunay stationary Gibbs states based on nonnegative interaction functions thanks to the use of the compactness of sublevel sets of the relative entropy.

Trash removal algorithm for fast construction of the elliptic Gabriel graph using Delaunay triangulation

Given a set of points P , finding near neighbors among the points is an important problem in many applications in CAD/CAM, computer graphics, computational geometry, etc. In this paper, we propose an efficient algorithm for constructing the elliptic Gabriel graph (EGG), which is a generalization of the well-known Gabriel graph and parameterized by a non-negative value α . Our algorithm is based on the observation that a candidate point which may define an edge of an EGG with a given point p ∈ P is always in the scaled Voronoi region of p with a scale factor 2 / α 2 when the parameter α < 1 . From this observation, we can reduce the search space for locating neighbors of p in the EGG. For the case of α ≥ 1 , due to the fact that EGG is a subgraph of the Delaunay graph of P , EGG can be efficiently computed by watching the validity of each edge in the Delaunay graph. The proposed algorithm is shown to have its time complexity as O ( n 3 ) in the worst case and O ( n ) in the average case when α is moderately close to unity. The idea presented in this paper may similarly apply to other problems for the proximity search for point sets.

A block LU-SGS implicit unsteady incompressible flow solver on hybrid dynamic grids for 2D external bio-fluid simulations

A hybrid dynamic grid generation technique for two-dimensional (2D) morphing bodies and a block lower–upper symmetric Gauss–Seidel (BLU-SGS) implicit dual-time-stepping method for unsteady incompressible flows are presented for external bio-fluid simulations. To discretize the complicated computational domain around 2D morphing configurations such as fishes and insect/bird wings, the initial grids are generated by a hybrid grid strategy firstly. Body-fitted quadrilateral (quad) grids are generated first near solid bodies. An adaptive Cartesian mesh is then generated to cover the entire computational domain. Cartesian cells which overlap the quad grids are removed from the computational domain, and a gap is produced between the quad grids and the adaptive Cartesian grid. Finally triangular grids are used to fill this gap. During the unsteady movement of morphing bodies, the dynamic grids are generated by a coupling strategy of the interpolation method based on ‘Delaunay graph’ and local remeshing technique. With the motion of moving/morphing bodies, the grids are deformed according to the motion of morphing body boundaries firstly with the interpolation strategy based on ‘Delaunay graph’ proposed by Liu and Qin. Then the quality of deformed grids is checked. If the grids become too skewed, or even intersect each other, the grids are regenerated locally. After the local remeshing, the flow solution is interpolated from the old to the new grid. Based on the hybrid dynamic grid technique, an efficient implicit finite volume solver is set up also to solve the unsteady incompressible flows for external bio-fluid dynamics. The fully implicit equation is solved using a dual-time-stepping approach, coupling with the artificial compressibility method (ACM) for incompressible flows. In order to accelerate the convergence history in each sub-iteration, a block lower–upper symmetric Gauss–Seidel implicit method is introduced also into the solver. The hybrid dynamic grid generator is tested by a group of cases of morphing bodies, while the implicit unsteady solver is validated by typical unsteady incompressible flow case, and the results demonstrate the accuracy and efficiency of present solver. Finally, some applications for fish swimming and insect wing flapping are carried out to demonstrate the ability for 2D external bio-fluid simulations.