On the vulnerabilities of voronoi-based approaches to mobile sensor deployment

Mobile sensor networks are the most promising solution to cover an Area of Interest (AoI) in safety critical scenarios. Mobile devices can coordinate with each other according to a distributed deployment algorithm, without resorting to human supervision for device positioning and network configuration. In this paper, we focus on the vulnerabilities of the deployment algorithms based on Voronoi diagrams to coordinate mobile sensors and guide their movements. We give a geometric characterization of possible attack configurations, proving that a simple attack consisting of a barrier of few compromised sensors can severely reduce network coverage. On the basis of the above characterization, we propose two new secure deployment algorithms, named SecureVor and Secure Swap Deployment (SSD). These algorithms allow a sensor to detect compromised nodes by analyzing their movements, under different and complementary operative settings. We show that the proposed algorithms are effective in defeating a barrier attack, and both have guaranteed termination. We perform extensive simulations to study the performance of the two algorithms and compare them with the original approach. Results show that SecureVor and SSD have better robustness and flexibility and excellent coverage capabilities and deployment time, even in the presence of an attac

Two problems in computational geometry

En aquesta tesi s'estudien dos problemes del camp de la geometria computacional. El primer problema és: donat un set S de n punts en el pla en posició general, com de prop són quatre punts de S de ser cocirculars. Definim tres mesures per estudiar aquesta qüestió, la mesura de Tales, la mesura de Voronoi, i la mesura del Determinant. Presentem cotes per la mesura de Tales, i algoritmes per computar aquestes mesures de cocircularitat. També reduïm el problema de computar la cocircularitat emprant la mesura del Determinant al problema de 4SUM. El segon problema és: donat dos sets R i B de punts rojos i blaus respectivament, com computar la discrepància bicromàtica amb caixes i cercles. La discrepància bicromàtica és definida com la diferència entre el nombre de punts vermells i blaus que són a l'interior de la figura examinada. Presentem una comparativa entre algoritmes ja existents per les dues figures. També comparem la discrepància bicromàtica de caixes orientades en els eixos vs. d'orientació general. A més a més, també presentem un nou algoritme per la discrepància en esferes/discs per a altes dimensions, basat en literatura ja existent. També relacionem altres problemes en el tema de separabilitat amb algoritmes sensitius a l'output per la discrepància amb caixes.En esta tesis se estudian dos problemas del campo de la geometría computacional. El primer problema es: dado un set S de n puntos en el plan en posición general, como de cerca son cuatro puntos de S de ser cocirculares. Definimos tres medidas para estudiar esta cuestión, la medida de Tales, la medida de Voronoi, y la medida del Determinante. Presentamos cotas por la medida de Tales, y algoritmos para computar estas medidas de cocircularidad. También reducimos el problema de computar la cocircularidad usando la medida del Determinante al problema de 4SUM. El segundo problema es: dado dos sets R y B de puntos rojos y azules respectivamente, como computar la discrepancia bicromática con cajas y círculos. La discrepancia bicromática es definida como la diferencia entre el número de puntos rojos y azules que están en el interior de la figura examinada. Presentamos una comparativa entre algoritmos ya existentes por las dos figuras. También comparamos la discrepancia bicromática de cajas orientadas en los ejes vs. de orientación general. Además, también presentamos un nuevo algoritmo por la discrepancia en esferas/discos para altas dimensiones, basado en literatura ya existente. También relacionamos otros problemas en el tema de separabilidad con algoritmos sensitivos al output por la discrepancia con cajas.Two different problems belonging to computational geometry are studied in this thesis. The first problem studies: given a set S of n points in the plane in general position, how close are four points of S to being cocircular. We define three measures to study this question, the Thales, Voronoi and Determinant measures. We present bounds on the Thales almost-cocircularity measure over a point set. Algorithms for computing these measures of cocircularity are presented as well. We give a reduction from computing cocircularity using the Determinant measure to the 4SUM problem. The second problem studies: given two sets R and B of red and blue points respectively, how to compute the bichromatic discrepancy using boxes and circles. The bichromatic discrepancy is defined as the difference between the number of red points and blue points inside the shape. We present a comparison of algorithms in the existing literature for the two shapes. Bichromatic discrepancy in axis-parallel boxes .vs non-axis-parallel boxes is also compared. Furthermore, we also present a new algorithm for disk discrepancy in higher dimensions, based on existing literature. We also relate existing problems in separability with existing output sensitive algorithms for bichromatic discrepancy using boxes

Aspects of Unstructured Grids and Finite-Volume Solvers for the Euler and Navier-Stokes Equations

One of the major achievements in engineering science has been the development of computer algorithms for solving nonlinear differential equations such as the Navier-Stokes equations. In the past, limited computer resources have motivated the development of efficient numerical schemes in computational fluid dynamics (CFD) utilizing structured meshes. The use of structured meshes greatly simplifies the implementation of CFD algorithms on conventional computers. Unstructured grids on the other hand offer an alternative to modeling complex geometries. Unstructured meshes have irregular connectivity and usually contain combinations of triangles, quadrilaterals, tetrahedra, and hexahedra. The generation and use of unstructured grids poses new challenges in CFD. The purpose of this note is to present recent developments in the unstructured grid generation and flow solution technology

Districting Problems - New Geometrically Motivated Approaches

This thesis focuses on districting problems were the basic areas are represented by points or lines. In the context of points, it presents approaches that utilize the problem\u27s underlying geometrical information. For lines it introduces an algorithm combining features of geometric approaches, tabu search, and adaptive randomized neighborhood search that includes the routing distances explicitly. Moreover, this thesis summarizes, compares and enhances existing compactness measures

Continuous Spatial Query Processing:A Survey of Safe Region Based Techniques

In the past decade, positioning system-enabled devices such as smartphones have become most prevalent. This functionality brings the increasing popularity of location-based services in business as well as daily applications such as navigation, targeted advertising, and location-based social networking. Continuous spatial queries serve as a building block for location-based services. As an example, an Uber driver may want to be kept aware of the nearest customers or service stations. Continuous spatial queries require updates to the query result as the query or data objects are moving. This poses challenges to the query efficiency, which is crucial to the user experience of a service. A large number of approaches address this efficiency issue using the concept of safe region . A safe region is a region within which arbitrary movement of an object leaves the query result unchanged. Such a region helps reduce the frequency of query result update and hence improves query efficiency. As a result, safe region-based approaches have been popular for processing various types of continuous spatial queries. Safe regions have interesting theoretical properties and are worth in-depth analysis. We provide a comparative study of safe region-based approaches. We describe how safe regions are computed for different types of continuous spatial queries, showing how they improve query efficiency. We compare the different safe region-based approaches and discuss possible further improvements

Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere

We present two exact implementations of efficient output-sensitive algorithms that compute Minkowski sums of two convex polyhedra in 3D. We do not assume general position. Namely, we handle degenerate input, and produce exact results. We provide a tight bound on the exact maximum complexity of Minkowski sums of polytopes in 3D in terms of the number of facets of the summand polytopes. The algorithms employ variants of a data structure that represents arrangements embedded on two-dimensional parametric surfaces in 3D, and they make use of many operations applied to arrangements in these representations. We have developed software components that support the arrangement data-structure variants and the operations applied to them. These software components are generic, as they can be instantiated with any number type. However, our algorithms require only (exact) rational arithmetic. These software components together with exact rational-arithmetic enable a robust, efficient, and elegant implementation of the Minkowski-sum constructions and the related applications. These software components are provided through a package of the Computational Geometry Algorithm Library (CGAL) called Arrangement_on_surface_2. We also present exact implementations of other applications that exploit arrangements of arcs of great circles embedded on the sphere. We use them as basic blocks in an exact implementation of an efficient algorithm that partitions an assembly of polyhedra in 3D with two hands using infinite translations. This application distinctly shows the importance of exact computation, as imprecise computation might result with dismissal of valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages long. The advisor was Prof. Dan Halperi

Limits of Voronoi Diagrams

In this thesis we study sets of points in the plane and their Voronoi diagrams, in particular when the points coincide. We bring together two ways of studying point sets that have received a lot of attention in recent years: Voronoi diagrams and compactifications of configuration spaces. We study moving and colliding points and this enables us to introduce `limit Voronoi diagrams'. We define several compactifications by considering geometric properties of pairs and triples of points. In this way we are able to define a smooth, real version of the Fulton-MacPherson compactification. We show how to define Voronoi diagrams on elements of these compactifications and describe the connection with the limit Voronoi diagrams.Comment: PhD thesis, 132 pages, lots of figure

Higher-order Voronoi diagrams of polygonal objects

Higher-order Voronoi diagrams are fundamental geometric structures which encode the k-nearest neighbor information. Thus, they aid in computations that require proximity information beyond the nearest neighbor. They are related to various favorite structures in computational geometry and are a fascinating combinatorial problem to study. While higher-order Voronoi diagrams of points have been studied a lot, they have not been considered for other types of sites. Points lack dimensionality which makes them unable to represent various real-life instances. Points are the simplest kind of geometric object and therefore higher- order Voronoi diagrams of points can be considered as the corner case of all higher-order Voronoi diagrams. The goal of this dissertation is to move away from the corner and bring the higher-order Voronoi diagram to more general geometric instances. We focus on certain polygonal objects as they provide flexibility and are able to represent real-life instances. Before this dissertation, higher-order Voronoi diagrams of polygonal objects had been studied only for the nearest neighbor and farthest Voronoi diagrams. In this dissertation we investigate structural and combinatorial properties and discover that the dimensionality of geometric objects manifests itself in numerous ways which do not exist in the case of points. We prove that the structural complexity of the order-k Voronoi diagram of non-crossing line segments is O(k(n-k)), as in the case of points. We study disjoint line segments, intersecting line segments, line segments forming a planar straight-line graph and extend the results to the Lp metric, 1<=p<=infty. We also establish the connection between two mathematical abstractions: abstract Voronoi diagrams and the Clarkson-Shor framework. We design several construction algorithms that cover the case of non-point sites. While computational geometry provides several approaches to study the structural complexity that give tight realizable bounds, developing an effective construction algorithm is still a challenging problem even for points. Most of the construction algorithms are designed to work with points as they utilize their simplicity and relations with data-structures that work specifically for points. We extend the iterative and the sweepline approaches that are quite efficient in constructing all order-i Voronoi diagrams, for i<=k and we also give three randomized construction algorithms for abstract higher-order Voronoi diagrams that deal specifically with the construction of the order-k Voronoi diagrams