Voronoi Vertex Research Articles

Energy aware optimal routing model for wireless multimedia sensor networks using modified Voronoi assisted prioritized double deep Q‐learning

SummaryThe energy hole problem is critical in Wireless Multimedia Sensor Network (WMSN), wherein the nodes closer to the sink node will expire sooner than the outer sub‐regions. Because they transmit their packet and forward the outer sub‐regions packet to the sink. Hence, a hole arises near the sink node after a very short time. Thus, an energy‐aware optimal routing model is introduced in this research using Modified Voronoi‐assisted prioritized double deep Q‐learning (PDDQL). Initially, the sensor nodes are deployed using the Voronoi cell structure that splits the entire region of interest into different Voronoi polygons. However, the Voronoi diagram will not always predict an efficient route because of the nodes at a Voronoi edge or vertex outside the transmission range. To tackle this issue, a new Voronoi diagram is proposed that depends on the maximum horizontal transmission range to remove the portion of Voronoi edges uncovered by the sensor node. After the deployment, a PDDQL algorithm is used to select the Relay Nodes based on their remaining energy. The proposed PDDQL determines an alternate route to transmit the packet if an energy hole problem has occurred in WMSN. The analysis of a proposed routing protocol based on the assessment measures like packet delivery ratio, average residual energy, number of alive nodes, and average end‐to‐end delay accomplished the value of 198.33 (ms), 2.05 (J), 100, and 0.85, respectively.

Energy-Efficient UAV Routing for Wireless Sensor Networks

Recently, an unmanned aerial vehicle (UAV) has been widely adopted to make efficient use of network resources in such areas as internet of things (IoT), sensor networks and three dimensional (3D) wireless networks. Especially, in wireless sensor networks (WSNs) where energy consumption of sensors in data transmission is the most conspicuous feature, data collection by UAV provides a promising solution. To address this issue, we consider a UAV-enabled WSN, where a UAV is dispatched to collect data from sensors distributed in networks. We formulate an optimization problem to maximize the minimum residual energy of sensors after data transmission for energy-efficient UAV routing subject to data collection and UAV traveling distance constraints. To solve the non-convex optimization problem, we first derive a feasible solution, i.e., the shortest UAV route that guarantees data collection at all the sensors, where a Voronoi diagram is modified to find a set of UAV hovering locations. The proposed algorithm preferentially determines each UAV hovering location at Voronoi vertex so that UAV can collect data from as many adjacent sensors as possible. Then with an initial shortest UAV route, a UAV route is proposed by adjusting each UAV hovering location sequentially based on sensor energy status, which is easily accomplished by the properties of Voronoi diagram. Lastly, to find the proposed solution more quickly, we propose a sensor-energy based initial UAV route determination method. Simulation results are provided to validate the performance of our proposed algorithm, and to compare with other UAV route determination schemes.

A Nearly Optimal Algorithm for the Geodesic Voronoi Diagram of Points in a Simple Polygon

The geodesic Voronoi diagram of m point sites inside a simple polygon of n vertices is a subdivision of the polygon into m cells, one to each site, such that all points in a cell share the same nearest site under the geodesic distance. The best known lower bound for the construction time is $$\varOmega (n+m\log m)$$, and a matching upper bound is a long-standing open question. The state-of-the-art construction algorithms achieve $$O( (n+m) \log (n+m) )$$ and $$O(n+m\log m\log ^2n)$$ time, which are optimal for $$m=\varOmega (n)$$ and $$m=O(\frac{n}{\log ^3n})$$, respectively. In this paper, we give a construction algorithm with $$O( n + m ( \log m+ \log ^2 n ) )$$ time, and it is nearly optimal in the sense that if a single Voronoi vertex can be computed in $$O(\log n)$$ time, then the construction time will become the optimal $$O(n+m\log m)$$. In other words, we reduce the problem of constructing the diagram in the optimal time to the problem of computing a single Voronoi vertex in $$O(\log n)$$ time.

Edge topology construction of Voronoi diagrams of spheres in non-general position

Although 3D Voronoi diagrams and medial axis transforms have numerous applications in biology, robotics, and manufacturing, most researchers use Voronoi diagrams of points instead of the true 3D input geometry, due to issues of robustness and scalability. In this paper, we present a robust sample-based GPU algorithm for calculating the full topology of Voronoi diagrams of non-general position spheres. Prior work demonstrated that the presence, geometry, and combinatorial basis of spheres that contribute to Voronoi vertices can be efficiently computed by shooting rays from each input sphere, mapping ray intersections with the nearest bisector surface to parametric bounding cubes, and analyzing the results. In this paper, we propose an algorithm on this parametric bounding cube to compute Voronoi edges in addition to the vertices. We successfully extract the full topology of the Voronoi diagram, including special cases such as isolated Voronoi edges that do not contain Voronoi vertices, more than three Voronoi edges emanating from a Voronoi vertex, and Voronoi edges that are shared by more than three Voronoi cells. Our GPU implementation efficiently and robustly handles all input, whether in general or non-general position, and finds all Voronoi vertices and edges, modulo the sampling density, including isolated disconnected edges.

SIR Meta Distribution for Spatiotemporal Cooperation in Poisson Cellular Networks

The meta-distribution provides more fine-grained information on the signal-to-interference ratio (SIR) compared to the SIR distribution at the typical user. This paper studies the SIR meta-distribution in heterogeneous cellular networks (HCNs) for JT-DPB with two types of hybrid automatic repeat request (HARQ) techniques, where JT-DPB is a general scheme for base station cooperation that combines joint transmission (JT) and dynamic point blanking (DPB). This paper also considers two types of typical users—the general user and the worst-case user (the typical user located at the Voronoi vertex in a single-tier network). For the Type-II HARQ, a simplified network model is proposed, which leads to a tractable and accurate approach for the SIR meta-distribution analysis. The analytical results for the Type-I HARQ and Type-II HARQ are presented.

Incremental Labelling of Voronoi Vertices for Shape Reconstruction

AbstractWe present an incremental Voronoi vertex labelling algorithm for approximating contours, medial axes and dominant points (high curvature points) from 2D point sets. Though there exist many number of algorithms for reconstructing curves, medial axes or dominant points, a unified framework capable of approximating all the three in one place from points is missing in the literature. Our algorithm estimates the normals at each sample point through poles (farthest Voronoi vertices of a sample point) and uses the estimated normals and the corresponding tangents to determine the spatial locations (inner or outer) of the Voronoi vertices with respect to the original curve. The vertex classification helps to construct a piece‐wise linear approximation to the object boundary. We provide a theoretical analysis of the algorithm for points non‐uniformly (ε‐sampling) sampled from simple, closed, concave and smooth curves. The proposed framework has been thoroughly evaluated for its usefulness using various test data. Results indicate that even sparsely and non‐uniformly sampled curves with outliers or collection of curves are faithfully reconstructed by the proposed algorithm.

The SIR Meta Distribution in Poisson Cellular Networks With Base Station Cooperation

The meta distribution provides fine-grained information on the signal-to-interference ratio (SIR) compared with the SIR distribution at the typical user. This paper first derives the meta distribution of the SIR in heterogeneous cellular networks with downlink coordinated multipoint transmission/reception, including joint transmission (JT), dynamic point blanking (DPB), and dynamic point selection/dynamic point blanking (DPS/DPB), for the general typical user and the worst-case user (the typical user located at the Voronoi vertex in a single-tier network). A more general scheme called JT-DPB, which is the combination of JT and DPB, is studied. The moments of the conditional success probability are derived for the calculation of the meta distribution and the mean local delay. An exact analytical expression, the beta approximation, and simulation results of the meta distribution are provided. From the theoretical results, we gain insights on the benefits of different cooperation schemes and the impact of the number of cooperating base stations and other network parameters.

Advances in co-volume mesh generation and mesh optimisation techniques

This paper introduces developments in modified techniques for the generation of unstructured, non-uniform, dual orthogonal meshes which are suitable for use with co-volume solution schemes. Two new mesh generation techniques, a modified advancing front technique and an octree-Delaunay algorithm, are coupled with a mesh optimisation algorithm. When using a Delaunay–Voronoi dual, to construct mutually orthogonal meshes for co-volume schemes, it is essential to minimise the number of Delaunay elements which do not contain their Voronoi vertex. These new techniques provide an improvement over previous approaches, as they produce meshes in which the number of elements that do not contain their Voronoi vertex is reduced. In particular, it is found that the optimisation algorithm, which could be applied to any mesh cosmetics problem, is very effective, regardless of the quality of the initial mesh. This is illustrated by applying the proposed approach to a number of complex industrial aerospace geometries.

Biometric security system over finite field for mobile applications

Recently, number of mobile users and mobile banking applications are increased dramatically. The current mobile banking system unable to provide face-to-face transaction, that is, anyone can perform transaction by using proper secret code. This study proposes a new triangular based indexing method to authenticate the original account holder in mobile banking environment. A Delaunay triangulation indexing scheme has been designed for protecting biometric template and Voronoi vertex is calculated for verifying the authorised person. A novel algorithm has been developed for mutual authentication. To reduce key size, elliptic curve cryptography technique is adapted. Biometric key is generated for secure communication and also discusses the security aspects of the proposed method.

Voronoi, Genetic Algorithms and Their Tandem Application in Wireless Sensor Network Deployment

Wireless Sensor Network (WSN) had become almost an indispensible especially the demand for data acquisition from national security to disaster mitigation management, weather data to environmental changes and from many more agencies. The effectiveness and efficacy of WSN dependent on the strength and weakness of the deployment of the sensor nodes which collect and transmit the data. The success of data acquisition in any network depended upon the adequacy of coverage by the sensor nodes; which in turn depended on the method of deployment and redeployment. Since deterministic deployment of nodes could not always be done, random deployment was adopted as a compulsion rather than an option. The random deployment of sensors by nature provided poor network coverage and leading to unsatisfactory data acquisition. Therefore, a better method was sought-after to redeploy the sensors that were deployed earlier at random. Hence, the compelling need had resulted in the development of numerous algorithms for suitably moving the sensors for maximum coverage. Such algorithms were of standalone ones or hybrid/combination in nature. One such combination algorithm termed as Voronoi-Genetic Algorithm (V-GA) a combination/tandom application of Voronoi Vertex Averaging Algorithm (VVAA) and Genetic Algorithm (GA) was analyzed in this study. The displacement and coverage performance were studied, analyzed and compared with that of random deployment and redeployment by the earlier proposed algorithms namely VVAA and GA by the same researcher.

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.

2D Centroidal Voronoi Tessellations with Constraints

We tackle the problem of constructing 2D centroidal Voronoi tessellations with constraints through an efficient and robust construction of bounded Voronoi diagrams, the pseudo-dual of the constrained Delaunay triangulation. We exploit the fact that the cells of the bounded Voronoi diagram can be obtained by clipping the ordinary ones against the constrained Delaunay edges. The clipping itself is efficiently computed by identifying for each constrained edge the (connected) set of triangles whose dual Voronoi vertex is hidden by the constraint. The resulting construction is amenable to Lloyd relaxation so as to obtain a centroidal tessellation with constraints.

Structure of void space in polymer solutions

The structure of void space in two- and three-dimensional (3D) polymer solutions is studied using Voronoi tessellation and percolation theory. The polymer molecules are modeled as freely jointed chains of N tangent hard disks (two dimensions) or spheres (three dimensions). Polymer chains are equilibrated via Monte Carlo simulations and the pore space in configurations of equilibrated chains is mapped using Voronoi tessellation. In d dimensions a Voronoi vertex is the center of the sphere tangent to the d+1 nearest monomers. An edge of the Voronoi diagram is the shortest route between two neighboring vertices. The edge is considered connected if a monomer can pass through and disconnected otherwise. The Voronoi construction is used to calculate the percolation threshold of the void space. The most interesting result is that the polymer area fraction at the percolation threshold is a nonmonotonic function of N in two dimensions but monotonically reaches a constant value in three dimensions. The crossover behavior of the percolation threshold is also observed in pseudo-3D. The pore size distribution decreases monotonically with increasing pore size. This is markedly different from that in configurations of hard disks (monomeric fluid) where the pore size distribution is peaked at finite size.

Voronoi diagram of a circle set from Voronoi diagram of a point set: II. Geometry

Presented in this paper are algorithms to compute the positions of vertices and equations of edges of the Voronoi diagram of a circle set on a plane, where the radii of the circles are not necessarily equal and the circles are not necessarily disjoint. The algorithms correctly and efficiently work in conjunction with the first paper of the series dealing with the construction of the correct topology of the Voronoi diagram of a circle set from the topology of the Voronoi diagram of a point set, where the points are centers of the circles. Given three circle generators, the position of the Voronoi vertex is computed by treating the plane as a complex plane, Z-plane, and transforming it into another complex plane, W-plane, via a linear fractional transformation. Then, the problem is formulated as a simple point location problem in regions defined by two lines and two circles in the W-plane. After the correct topology is constructed with the geometry of the vertices, the equations of edge are computed in a rational quadratic Bézier curve from.

Hunting Voronoi vertices

Given three sites in the plane, a Voronoi vertex is a point that is equidistant from each. In this paper, we consider the problem of computing Voronoi vertices for three disjoint planar sites of fixed but possibly unknown shape; we only require the ability to query the closest point on an object from a given point. Each site is assumed to be either convex or a finite union of convex sets. Our technique is simple and iterative in nature: beginning from some initial seed point, it computes a sequence of points based on intermediate closest-point queries. This technique is observed to either converge to a Voronoi vertex or oscillate with some finite period. We study geometric conditions on shape/placement of the objects and choice of the initial point that guarantee convergence or oscillation. We show that our technique is probabilistically complete: selecting seed points at random will guarantee convergence to a finite Voronoi vertex, if one exists. Our motivation for seeking Voronoi vertices comes from robot motion planning: Voronoi vertices are natural havens for mobile robots avoiding obstacles. We conclude by briefly describing the implementation of a retraction-like path planner for a planar robot based on our iterative strategy for seeking Voronoi vertices.

A hashing-oriented nearest neighbor searching scheme

This paper presents a hashing-oriented nearest neighbor searching scheme. Given n points in the Euclidean two-dimensional plane, we first construct a Voronoi diagram and record the Voronoi vertices and the Voronoi edges. By passing each Voronoi vertex, we use two perpendicular lines, one is horizontal and the other is vertical, to partition the plane into some rectangular subdivisions. Here, each rectangular subdivision dominates at most two given points. Then we establish two hashing functions corresponding to horizontal slabs and vertical slabs, respectively. By applying the established hashing functions, we can quickly determine the proper rectangular subdivision containing the query point. After that, we compare the distance between the query point and the dominated points to determine the nearest neighbor. The searching time by our scheme is O(1). The preprocessing time and the amount of required storage are O( n 2 + t), respectively, where n is the number of given points and t is the size of the hashing table needed by the established hashing functions.