Increasing numbers of people are subscribing to location-based services, but as the popularity grows so are the privacy concerns. Varieties of research exist to address these privacy concerns. Each technique tries to address different models with which location-based services respond to subscribers. In this work, we present ideas to address privacy concerns for the two main models namely: the snapshot nearest neighbor query model and the continuous nearest neighbor query model. First, we address snapshot nearest neighbor query model where location-based services response represents a snapshot of point in time. In this model, we introduce a novel idea based on the concept of an open set in a topological space where points belongs to a subset called neighborhood of a point. We extend this concept to provide anonymity to real objects where each object belongs to a disjointed neighborhood such that each neighborhood contains a single object. To help identify the objects, we implement a database which dynamically scales in direct proportion with the size of the neighborhood. To retrieve information secretly and allow the database to expose only requested information, private information retrieval protocols are executed twice on the data. Our study of the implementation shows that the concept of a single object neighborhood is able to efficiently scale the database with the objects in the area. The size of the database grows with the size of the grid and the objects covered by the location-based services. Typically, creating neighborhoods, computing distances between objects in the area, and running private information retrieval protocols causes the CPU to respond slowly with this increase in database size. In order to handle a large number of objects, we explore the concept of kernel and parallel computing in GPU. We develop GPU parallel implementation of the snapshot query to handle large number of objects. In our experiment, we exploit parameter tuning. The results show that with parameter tuning and parallel computing power of GPU we are able to significantly reduce the response time as the number of objects increases. To determine response time of an application without knowledge of the intricacies of GPU architecture, we extend our analysis to predict GPU execution time. We develop the run time equation for an operation and extrapolate the run time for a problem set based on the equation, and then we provide a model to predict GPU response time. As an alternative, the snapshot nearest neighbor query privacy problem can be addressed using secure hardware computing which can eliminate the need for protecting the rest of the sub-system, minimize resource usage and network transmission time. In this approach, a secure coprocessor is used to provide privacy. We process all information inside the coprocessor to deny adversaries access to any private information. To obfuscate access pattern to external memory location, we use oblivious random access memory methodology to access the server. Experimental evaluation shows that using a secure coprocessor reduces resource usage and query response time as the size of the coverage area and objects increases. Second, we address privacy concerns in the continuous nearest neighbor query model where location-based services automatically respond to a change in object*s location. In this model, we present solutions for two different types known as moving query static object and moving query moving object. For the solutions, we propose plane partition using a Voronoi diagram, and a continuous fractal space filling curve using a Hilbert curve order to create a continuous nearest neighbor relationship between the points of interest in a path. Specifically, space filling curve results in multi-dimensional to 1-dimensional object mapping where values are assigned to the objects based on proximity. To prevent subscribers from issuing a query each time there is a change in location and to reduce the response time, we introduce the concept of transition and update time to indicate where and when the nearest neighbor changes. We also introduce a database that dynamically scales with the size of the objects in a path to help obscure and relate objects. By executing the private information retrieval protocol twice on the data, the user secretly retrieves requested information from the database. The results of our experiment show that using plane partitioning and a fractal space filling curve to create nearest neighbor relationships with transition time between objects reduces the total response time.

<p>Entering the era of big data, human beings are faced with an unprecedented amount of geometric data today. Many computational challenges arise in processing the new deluge of geometric data. A critical one is data uncertainty: the data is inherently noisy and inaccuracy, and often lacks of completeness. The past few decades have witnessed the influence of geometric algorithms in various fields including GIS, spatial databases, and computer vision, etc. Yet most of the existing geometric algorithms are built on the assumption of the data being precise and are incapable of properly handling data in the presence of uncertainty. This thesis explores a few algorithmic challenges in what we call geometric computing over uncertain data.</p><p>We study the nearest-neighbor searching problem, which returns the nearest neighbor of a query point in a set of points, in a probabilistic framework. This thesis investigates two different nearest-neighbor formulations: expected nearest neighbor (ENN), where we consider the expected distance between each input point and a query point, and probabilistic nearest neighbor (PNN), where we estimate the probability of each input point being the nearest neighbor of a query point.</p><p>For the ENN problem, we consider a probabilistic framework in which the location of each input point and/or query point is specified as a probability density function and the goal is to return the point that minimizes the expected distance. We present methods for computing an exact ENN or an \\eps-approximate ENN, for a given error parameter 0 < \\eps < 1, under different distance functions. These methods build an index of near-linear size and answer ENN queries in polylogarithmic or sublinear time, depending on the underlying function. As far as we know, these are the first nontrivial methods for answering exact or \\eps-approximate ENN queries with provable performance guarantees. Moreover, we extend our results to answer exact or \\eps-approximate k-ENN queries. Notably, when only the query points are uncertain, we obtain state-of-the-art results for top-k aggregate (group) nearest-neighbor queries in the L1 metric using the weighted SUM operator.</p><p>For the PNN problem, we consider a probabilistic framework in which the location of each input point is specified as a probability distribution function. We present efficient algorithms for (i) computing all points that are nearest neighbors of a query point with nonzero probability; (ii) estimating, within a specified additive error, the probability of a point being the nearest neighbor of a query point; (iii) using it to return the point that maximizes the probability being the nearest neighbor, or all the points with probabilities greater than some threshold to be the nearest neighbor. We also present some experimental results to demonstrate the effectiveness of our approach.</p><p>We study the convex-hull problem, which asks for the smallest convex set that contains a given point set, in a probabilistic setting. In our framework, the uncertainty of each input point is described by a probability distribution over a finite number of possible locations including a null location to account for non-existence of the point. Our results include both exact and approximation algorithms for computing the probability of a query point lying inside the convex hull of the input, time-space tradeoffs for the membership queries, a connection between Tukey depth and membership queries, as well as a new notion of \\beta-hull that may be a useful representation of uncertain hulls.</p><p>We study contour trees of terrains, which encode the topological changes of the level set of the height value \\ell as we raise \\ell from -\\infty to +\\infty on the terrains, in a probabilistic setting. We consider a terrain that is defined by linearly interpolating each triangle of a triangulation. In our framework, the uncertainty lies in the height of each vertex in the triangulation, and we assume that it is described by a probability distribution. We first show that the probability of a vertex being a critical point, and the expected number of nodes (resp. edges) of the contour tree, can be computed exactly efficiently. Then we present efficient sampling-based methods for estimating, with high probability, (i) the probability that two points lie on an edge of the contour tree, within additive error; (ii) the expected distance of two points p, q and the probability that the distance of p, q is at least \\ell on the contour tree, within additive error and/or relative error, where the distance of p, q on a contour tree is defined to be the difference between the maximum height and the minimum height on the unique path from p to q on the contour tree.</p> / Dissertation

In the future, more data intensive applications, such as P2P auction networks, P2P job--search networks, P2P multi--player games, will require the capability to respond to more complex queries such as the nearest neighbor queries involving numerous data types. For the problem of answering nearest neighbor queries (NN query) for spatial region data in the P2P environment, a quadtree-based structure probably is a good choice. However, the quadtree stores the data in the leaf nodes, resulting in the load unbalance and expensive cost of any query. The MX--CIF quadtree can solve this problem. The MX--CIF quadtree has three properties: controlling efficiently the height of the tree, reducing load unbalance, and reducing the NNquery scope with controlling the value of the radius. Although the P2P MX--CIF quadtree can do the NN query efficiently, it still has some problems as follows: low accuracy of the nearest neighbor query, the expensive cost of the tree construction, the high search cost of the NN query, and load unbalance. In fact, the index structures for the region data can also work for the point data which can be considered as the degenerated case of the region data. Therefore, the KDB--tree which is a well-known algorithm for the point data can be used to reduce load unbalance, but it has the same problem as the quadtree. The data is stored only in the leaf nodes of the KDB--tree. In this thesis, we propose an Adjustable KDB--tree (AKDB--tree) to improve this situation for the P2P system. The AKDB--tree has five properties: reducing load unbalance, low cost of the tree construction, storing the data in the internal nodes and leaf nodes, high accuracy and low search cost of the NN query. The Chord system is a well--known structured P2P system in which the data search is performed by a hash function, instead of flooding used in most of the unstructured P2P system. Since the Chord system is a hash approach, it is easy to deal with peers joining/exiting. Besides, in order to combine AKDB--tree with the Chord system, we design the IDs of the nodes in the AKDB--tree. Each node is hashed to the Chord system by the ID. The IDs can be used to differentiate the edge node in the AKDB-tree is a vertical edge or a horizontal edge and the relative position of two nodes in the 2D space. And, we can calculate the related edge of a region in the 2D space according to the ID of the region. As discussed above, we make use of the property of IDs to reduce the search cost of the NN query by a wide margin. In our simulation study, we compare our method with the P2P MX--CIF quadtree by considering five performance measures under four different situations of the P2P MX--CIF quadtree. From our simulation results, for the NN query, our AKDB-tree can provide the higher accuracy and lower search cost than the P2P MX--CIF quadtree. For the problem of load, our AKDB-tree is more balance than the P2P MX--CIF quadtree. For the time of the tree construction, our AKDB-tree needs shorter time than the P2P MX--CIF quadtree.

Ein zunehmend wichtiges Thema in der Informatik ist der Umgang mit Ähnlichkeit in einer großen Anzahl unterschiedlicher Domänen. Derzeit existiert keine universell verwendbare Infrastruktur für die Ähnlichkeitssuche in allgemeinen metrischen Räumen. Ziel der Arbeit ist es, die Grundlage für eine derartige Infrastruktur zu legen, die in klassische Datenbankmanagementsysteme integriert werden könnte. Im Rahmen einer Analyse des State of the Art wird der M-Baum als am besten geeignete Basisstruktur identifiziert. Dieser wird anschließend zum EM-Baum erweitert, wobei strukturelle Kompatibilität mit dem M-Baum erhalten wird. Die Abfragealgorithmen werden im Hinblick auf eine Minimierung notwendiger Distanzberechnungen optimiert. Aufbauend auf einer mathematischen Analyse der Beziehung zwischen Baumstruktur und Abfrageaufwand werden Freiheitsgrade in Baumänderungsalgorithmen genutzt, um Bäume so zu konstruieren, dass Ähnlichkeitsanfragen mit einer minimalen Anzahl an Anfrageoperationen beantwortet werden können. / A topic of growing importance in computer science is the handling of similarity in multiple heterogenous domains. Currently there is no common infrastructure to support this for the general metric space. The goal of this work is lay the foundation for such an infrastructure, which could be integrated into classical data base management systems. After some analysis of the state of the art the M-Tree is identified as most suitable base and enhanced in multiple ways to the EM-Tree retaining structural compatibility. The query algorithms are optimized to reduce the number of necessary distance calculations. On the basis of a mathematical analysis of the relation between the tree structure and the query performance degrees of freedom in the tree edit algorithms are used to build trees optimized for answering similarity queries using a minimal number of distance calculations.

Ein zunehmend wichtiges Thema in der Informatik ist der Umgang mit Ähnlichkeit in einer großen Anzahl unterschiedlicher Domänen. Derzeit existiert keine universell verwendbare Infrastruktur für die Ähnlichkeitssuche in allgemeinen metrischen Räumen. Ziel der Arbeit ist es, die Grundlage für eine derartige Infrastruktur zu legen, die in klassische Datenbankmanagementsysteme integriert werden könnte. Im Rahmen einer Analyse des State of the Art wird der M-Baum als am besten geeignete Basisstruktur identifiziert. Dieser wird anschließend zum EM-Baum erweitert, wobei strukturelle Kompatibilität mit dem M-Baum erhalten wird. Die Abfragealgorithmen werden im Hinblick auf eine Minimierung notwendiger Distanzberechnungen optimiert. Aufbauend auf einer mathematischen Analyse der Beziehung zwischen Baumstruktur und Abfrageaufwand werden Freiheitsgrade in Baumänderungsalgorithmen genutzt, um Bäume so zu konstruieren, dass Ähnlichkeitsanfragen mit einer minimalen Anzahl an Anfrageoperationen beantwortet werden können. / A topic of growing importance in computer science is the handling of similarity in multiple heterogenous domains. Currently there is no common infrastructure to support this for the general metric space. The goal of this work is lay the foundation for such an infrastructure, which could be integrated into classical data base management systems. After some analysis of the state of the art the M-Tree is identified as most suitable base and enhanced in multiple ways to the EM-Tree retaining structural compatibility. The query algorithms are optimized to reduce the number of necessary distance calculations. On the basis of a mathematical analysis of the relation between the tree structure and the query performance degrees of freedom in the tree edit algorithms are used to build trees optimized for answering similarity queries using a minimal number of distance calculations.