Delaunay Graph Research Articles

An Optimal Control Approach to Decentralized Energy-Efficient Coverage Problems

One of the challenges of working with multi-agent systems is the limited energy of the agents, especially when the agents are flying vehicles with limited batteries to save weight. The main contribution of this paper is to formulate the energy-efficient coverage problem as an optimal control problem. The optimal control problem will be related to Lloyd's algorithm. The solution to the optimal control problem is spatially distributed over Delaunay graphs and provides an energy-efficient local controller to maximize the coverage. As a second contribution, by imposing constraints on the parameters of the optimal control problem we guarantee that the agents maintain their energy during the coverage task. As expected, weighting the speed of the agents against the coverage objective will decrease energy consumption in the multi-agent system. Several examples demonstrate the performance of the energy-efficient approach.

Approximate nearest neighbor algorithm based on navigable small world graphs

We propose a novel approach to solving the approximate k-nearest neighbor search problem in metric spaces. The search structure is based on a navigable small world graph with vertices corresponding to the stored elements, edges to links between them, and a variation of greedy algorithm for searching. The navigable small world is created simply by keeping old Delaunay graph approximation links produced at the start of construction. The approach is very universal, defined in terms of arbitrary metric spaces and at the same time it is very simple. The algorithm handles insertions in the same way as queries: by finding approximate neighbors for the inserted element and connecting it to them. Both search and insertion can be done in parallel requiring only local information from the structure. The structure can be made distributed. The accuracy of the probabilistic k-nearest neighbor queries can be adjusted without rebuilding the structure.The performed simulation for data in the Euclidean spaces shows that the structure built using the proposed algorithm has small world navigation properties with log2(n) insertion and search complexity at fixed accuracy, and performs well at high dimensionality. Simulation on a CoPHiR dataset revealed its high efficiency in case of large datasets (more than an order of magnitude less metric computations at fixed recall) compared to permutation indexes. Only 0.03% of the 10 million 208-dimensional vector dataset is needed to be evaluated to achieve 0.999 recall (virtually exact search). For recall 0.93 processing speed 2800queries/s can be achieved on a dual Intel X5675 Xenon server node with Java implementation.

CD-graph: planar graph representation for spatial adjacency and neighbourhood relation with constraints

Neighbourhood relational graphs are widely used in Geosciences. Given a set of spatial objects (vertices) in the plane together with a set of spatial obstacles and spatial facilitators in straight-line edges, the constrained Delaunay graph (CD-graph) is an undirected graph representing the spatial adjacency and neighbourhood relation of objects. CD-graph is an approximated triangulation of vertices with the following properties: (1) the obstacles are included in the graph as some barrier edges that block the connection of the objects on both sides of the obstacles, and the facilitators are included in the graph as some nontrivial edges that connect the objects that are broken by the obstacles; (2) it is as close as possible to the Delaunay triangulation (D-TIN). CD-graph can be used to represent the spatial adjacency and neighbourhood relation of objects with constraints. A theoretical contrast is conducted to differentiate CD-graph, arbitrary (unconstrained) D-TIN and constrained D-TIN. Meanwhile, a two-step constraint-embedding algorithm is proposed to build CD-graph in optimal time by using divide-and-conquer technique. Subsequently, the Voronoi diagram and D-TIN-based k-order neighbours is extended in CD-graph to express different scales of spatial adjacency and neighbourhood relation of objects. CD-graph can be widely used in geographical applications, such as spatial interpolation, spatial clustering and spatial decision support.

Parallelized Aeroelastic Investigation Based on Delaunay Graph Mapping

Parallelized Aeroelastic Investigation Based on Delaunay Graph Mapping

Parallelized Aeroelastic Investigation Based on Delaunay Graph Mapping

In this paper, an effective and well robust dynamic grid deformation method based on Delaunay graph mapping is developed to solve the deformation of the 3-D hybrid multiblocks grids which is combined by near-wall viscosity grids and unstructured grids. Further more, the static aeroelastics problems of the standard model M6 elastic wing is investigated altogether by coupled with structure dynamic equation. The comparison, analysis and investigation were done as well. The CFD grids domain is subdivided into subdomains for parallel computation. And the program is carried out by MPI parallel computation standards.

Video key frame extraction through dynamic Delaunay clustering with a structural constraint

Key frame based video summarization has emerged as an important area of research for the multimedia community. Video key frames enable an user to access any video in a friendly and meaningful way. In this paper, we propose an automated method of video key frame extraction using dynamic Delaunay graph clustering via an iterative edge pruning strategy. A structural constraint in form of a lower limit on the deviation ratio of the graph vertices further improves the video summary. We also employ an information-theoretic pre-sampling where significant valleys in the mutual information profile of the successive frames in a video are used to capture more informative frames. Various video key frame visualization techniques for efficient video browsing and navigation purposes are incorporated. A comprehensive evaluation on 100 videos from the Open Video and YouTube databases using both objective and subjective measures demonstrate the superiority of our key frame extraction method.

The k-Nearest-Neighbor Voronoi Diagram Revisited

We revisit the k-nearest-neighbor (k-NN) Voronoi diagram and present a new paradigm for its construction. We introduce the k-NN Delaunay graph, which is the graph-theoretic dual of the k-NN Voronoi diagram, and use it as a base to directly compute this diagram in R 2. We implemented our paradigm in the L 1 and L ? metrics, using segment-dragging queries, resulting in the first output-sensitive, O((n+m)logn)-time algorithm to compute the k-NN Voronoi diagram of n points in the plane, where m is the structural complexity (size) of this diagram. We also show that the structural complexity of the k-NN Voronoi diagram in the L ? (equiv. L 1) metric is O(min{k(n?k),(n?k)2}). Efficient implementation of our paradigm in the L 2 (resp. L p , 1<p<?) metric remains an open problem.

Exact Voronoi diagram of smooth convex pseudo-circles: General predicates, and implementation for ellipses

We examine the problem of computing exactly the Voronoi diagram (via the dual Delaunay graph) of a set of, possibly intersecting, smooth convex pseudo-circles in the Euclidean plane, given in parametric form. Pseudo-circles are (convex) sites, every pair of which has at most two intersecting points. The Voronoi diagram is constructed incrementally. Our first contribution is to propose robust and efficient algorithms, under the exact computation paradigm, for all required predicates, thus generalizing earlier algorithms for non-intersecting ellipses. Second, we focus on InCircle, which is the hardest predicate, and express it by a simple sparse 5×5 polynomial system, which allows for an efficient implementation by means of successive Sylvester resultants and a new factorization lemma. The third contribution is our cgal-based c++ software for the case of possibly intersecting ellipses, which is the first exact implementation for the problem. Our code spends about a minute to construct the Voronoi diagram of 200 ellipses, when few degeneracies occur. It is faster than the cgal segment Voronoi diagram, when ellipses are approximated by k-gons for k>15, and a state-of-the-art implementation of the Voronoi diagram of points, when each ellipse is approximated by more than 1250 points.

A uniqueness theorem for Delaunay graphs

We establish a necessary and sufficient condition for the congruence of two isomorphic Delaunay graphs.

Aerodynamic design optimization of nacelle/pylon position on an aircraft

The arbitrary space-shape free form deformation (FFD) method developed in this paper is based on non-uniform rational B-splines (NURBS) basis function and used for the integral parameterization of nacelle-pylon geometry. The multi-block structured grid deformation technique is established by Delaunay graph mapping method. The optimization objects of aerodynamic characteristics are evaluated by solving Navier–Stokes equations on the basis of multi-block structured grid. The advanced particle swarm optimization (PSO) is utilized as search algorithm, which combines the Kriging model as surrogate model during optimization. The optimization system is used for optimizing the nacelle location of DLR-F6 wing-body-pylon-nacelle. The results indicate that the aerodynamic interference between the parts is significantly reduced. The optimization design system established in this paper has extensive applications and engineering value.

Exact Computation of the Topology and Geometric Invariants of the Voronoi Diagram of Spheres in 3D

In this paper, we are addressing the exact computation of the Delaunay graph (or quasi-triangulation) and the Voronoi diagram of spheres using Wu’s algorithm. Our main contributions are first a methodology for automated derivation of invariants of the Delaunay empty circumsphere predicate for spheres and the Voronoi vertex of four spheres, then the application of this methodology to get all geometrical invariants that intervene in this problem and the exact computation of the Delaunay graph and the Voronoi diagram of spheres. To the best of our knowledge, there does not exist a comprehensive treatment of the exact computation with geometrical invariants of the Delaunay graph and the Voronoi diagram of spheres. Starting from the system of equations defining the zero-dimensional algebraic set of the problem, we are applying Wu’s algorithm to transform the initial system into an equivalent Wu characteristic (triangular) set. In the corresponding system of algebraic equations, in each polynomial (except the first one), the variable with higher order from the preceding polynomial has been eliminated (by pseudo-remainder computations) and the last polynomial we obtain is a polynomial of a single variable. By regrouping all the formal coefficients for each monomial in each polynomial, we get polynomials that are invariants for the given problem. We rewrite the original system by replacing the invariant polynomials by new formal coefficients. We repeat the process until all the algebraic relationships (syzygies) between the invariants have been found by applying Wu's algorithm on the invariants. Finally, we present an incremental algorithm for the construction of Voronoi diagrams and Delaunay graphs of spheres in 3D and its application to Geodesy.

Investigation of Trailing-Edge Flap Gap Effects on Rotor Performance Using High-Fidelity Analysis

Effects of trailing-edge flap gaps on rotor performance are investigated using a high-fidelity coupled computational fluid dynamics computational structural dynamics analysis. Both integral flap (the flap is an integral part of the blade such that there are no physical gaps at the flap ends) and discrete flap (the flap is a separate entity with physical gaps in the span and chord directions) are examined on an UH-60A rotor at high-speed forward-flight conditions. A novel grid deformation scheme based on the Delaunay graph mapping is developed and implemented to allow the computational fluid dynamics modeling of the gaps with minimal distortion of mesh around the flap gap regions. This method offers an alternative to the traditional approach of modeling such configurations using overset meshes. The simulation results show that the effectiveness of the flap is minimally affected with span gaps; the penalty on rotor performance is of the order of 1% compared to the integral flap. On the other hand, the chord gaps significantly degrade the benefits of active flap on rotor performance due to the flow penetration between the upper and lower surfaces of the flap.

PROXIMITY GRAPHS: E, δ, Δ, χ AND ω

Graph-theoretic properties of certain proximity graphs defined on planar point sets are investigated. We first consider some of the most common proximity graphs of the family of the Delaunay graph, and study their number of edges, minimum and maximum degree, clique number, and chromatic number. In the second part of the paper we focus on the higher order versions of some of these graphs and give bounds on the same parameters.

Towards higher-dimensional topological self-stabilization: A distributed algorithm for Delaunay graphs

This article studies the construction of self-stabilizing topologies for distributed systems. While recent research has focused on chain topologies where nodes need to be linearized with respect to their identifiers, we explore a natural and relevant 2-dimensional generalization. In particular, we present a local self-stabilizing algorithm DStab which is based on the concept of “local Delaunay graphs” and which forwards temporary edges in greedy fashion reminiscent of compass routing. DStab constructs a Delaunay graph from any initial connected topology and in a distributed manner in time O(n3) in the worst-case; if the initial network contains the Delaunay graph, the convergence time is only O(n) rounds. DStab also ensures that individual node joins and leaves affect a small part of the network only. Such self-stabilizing Delaunay networks have interesting applications and our construction gives insights into the necessary geometric reasoning that is required for higher-dimensional linearization problems.

Applications of dynamic hybrid grid method for three-dimensional moving/deforming boundary problems

In the previous work, the authors had developed a dynamic hybrid grid generation method and an unsteady flow solver for two-dimensional incompressible and compressible unsteady flows with moving or morphing boundary. In this paper, the dynamic hybrid grid generation method and the unsteady flow solver are extended to three-dimensional complex geometries with moving and/or deforming boundaries, and coupled with force and moment calculation, and the integration of the rigid body, six degrees-of-freedom (6DOF) equations of motion. In order to enhance the flexibility and efficiency of moving grid generation, the dynamic hybrid grid method combines the ‘Delaunay graph’ mapping approach, node relaxation based on ‘spring’ analogy and local re-meshing strategy. Firstly, the prism/tetrahedral/Cartesian hybrid grids are adopted to discrete the initial computational domain over complex configurations. Once the bodies move or deform, the grid points in the boundary layer of the moving/morphing bodies are moved firstly with a modified advancing-layer method, the grid points in the outer far-field keep stationary, while the grid points between the last layer of body-fitted grids and the internal boundary of the specified far-field are mapped by the ‘Delaunay graph’ mapping method. But the background grids (the Delaunay graph) themselves are deformed by the simple node relaxation based on ‘spring’ analogy to improve the efficiency. Then the quality of the deformed grids is checked with some criteria. If the deformed grids do not pass the checking step, a local re-meshing procedure is carried out. Based on the dynamic hybrid grids, a parallel implicit finite-volume flow solver for 3D unsteady Navier–Stokes equations is developed also. In order to deal with the problems of multi-body separation, the integration of the rigid body, 6DOF equations of motion is coupled in the same framework of the flow solver. The applications for complex 3D morphing configurations demonstrate the robustness and efficiency of present method.

Gabriel Graphs in Arbitrary Metric Space and their Cellular Automaton for Many Grids

Gabriel graphs are subgraphs of Delaunay graphs that are used in many domains such as sensor networks and computer graphics. Although very useful in their original form, their definition is bounded to applications involving Euclidean spaces only, but their principles seem to be applicable to a wider range of applications. In this article, we generalize this construct and define metric Gabriel graphs that transport the principles of Gabriel graphs on arbitrary metric space, allowing their use in domains like cellular automata and amorphous computing, or any other domains where a non-Euclidean metric is used. We study global/local properties of metric Gabriel graphs and use them to design a cellular automaton that draws the metric Gabriel graph of its input. This cellular automaton only uses seven states to achieve this goal and has been tested on hexagonal grids, 4-connected, and 8-connected square grids.

A Partial Unit Delaunay Graph with Planar and Spanner for Ad Hoc Wireless Networks

This paper proposes a new geometry structure, namely Partial unit Delaunay graph (PuDel), to be as the underlying network topology of Ad hoc wireless networks. PuDel has the following attractive properties: (1) PuDel is a connected subgraph of unit Delaunay triangulation; (2) PuDel is a planar spanner of Unit Disk Graph (UDG) whose length stretch factor is at most 1.21; and (3) PuDel can be locally constructed only based on the position information of 1-hops neighbors, and requires no message exchange between nodes in addition to maintaining UDG.

Spanners of additively weighted point sets

We study the problem of computing geometric spanners for (additively) weighted point sets. A weighted point set is a set of pairs ( p , r ) where p is a point in the plane and r is a real number. The distance between two points ( p i , r i ) and ( p j , r j ) is defined as | p i p j | − r i − r j . We show that in the case where all r i are positive numbers and | p i p j | ⩾ r i + r j for all i, j (in which case the points can be seen as non-intersecting disks in the plane), a variant of the Yao graph is a ( 1 + ϵ ) -spanner that has a linear number of edges. We also show that the Additively Weighted Delaunay graph (the face-dual of the Additively Weighted Voronoi diagram) has a spanning ratio bounded by a constant. The straight-line embedding of the Additively Weighted Delaunay graph may not be a plane graph. Given the Additively Weighted Delaunay graph, we show how to compute a plane straight-line embedding that also has a spanning ratio bounded by a constant in O ( n log n ) time.

Delaunay Graph Based Path Planning of Dynamic Obstacle Avoidance Method for Mobile

This paper is about dynamic obstacle avoidance. Delaunay Graph is used for modeling the working space, an approximate shortest path of mobile robot is determined by using floyd algorithm. Path can be found easily with genetic algorithm. Then genetic algorithm is used for obtaining the optimum path. It may meet which dynamic obstacle when robot follows optimum path. so it should avoid it. Results of simulation show that this path planning method is simple and realized easily.

Topology-Based Hierarchical Clustering of Self-Organizing Maps

A powerful method in the analysis of datasets where there are many natural clusters with varying statistics such as different sizes, shapes, density distribution, overlaps, etc., is the use of self-organizing maps (SOMs). However, further processing tools, such as visualization and interactive clustering, are often necessary to capture the clusters from the learned SOM knowledge. A recent visualization scheme (CONNvis) and its interactive clustering utilize the data topology for SOM knowledge representation by using a connectivity matrix (a weighted Delaunay graph), CONN. In this paper, we propose an automated clustering method for SOMs, which is a hierarchical agglomerative clustering of CONN. We determine the number of clusters either by using cluster validity indices or by prior knowledge on the datasets. We show that, for the datasets used in this paper, data-topology-based hierarchical clustering can produce better partitioning than hierarchical clustering based solely on distance information.