Approximate Nearest Neighbor Search for Low Dimensional Queries

We study the Approximate Nearest Neighbor problem for metric spaces where the query points are constrained to lie on a subspace of low doubling dimension, while the data is high-dimensional. We show that this problem can be solved efficiently despite the high dimensionality of the data.Comment: 25 page

On trip planning queries in spatial databases

In this paper we discuss a new type of query in Spatial Databases, called Trip Planning Query (TPQ). Given a set of points P in space, where each point belongs to a category, and given two points s and e, TPQ asks for the best trip that starts at s, passes through exactly one point from each category, and ends at e. An example of a TPQ is when a user wants to visit a set of different places and at the same time minimize the total travelling cost, e.g. what is the shortest travelling plan for me to visit an automobile shop, a CVS pharmacy outlet, and a Best Buy shop along my trip from A to B? The trip planning query is an extension of the well-known TSP problem and therefore is NP-hard. The difficulty of this query lies in the existence of multiple choices for each category. In this paper, we first study fast approximation algorithms for the trip planning query in a metric space, assuming that the data set fits in main memory, and give the theory analysis of their approximation bounds. Then, the trip planning query is examined for data sets that do not fit in main memory and must be stored on disk. For the disk-resident data, we consider two cases. In one case, we assume that the points are located in Euclidean space and indexed with an Rtree. In the other case, we consider the problem of points that lie on the edges of a spatial network (e.g. road network) and the distance between two points is defined using the shortest distance over the network. Finally, we give an experimental evaluation of the proposed algorithms using synthetic data sets generated on real road networks

On trip planning queries in spatial databases

In this paper we discuss a new type of query in Spatial Databases, called Trip Planning Query (TPQ). Given a set of points P in space, where each point belongs to a category, and given two points s and e, TPQ asks for the best trip that starts at s, passes through exactly one point from each category, and ends at e. An example of a TPQ is when a user wants to visit a set of different places and at the same time minimize the total travelling cost, e.g. what is the shortest travelling plan for me to visit an automobile shop, a CVS pharmacy outlet, and a Best Buy shop along my trip from A to B? The trip planning query is an extension of the well-known TSP problem and therefore is NP-hard. The difficulty of this query lies in the existence of multiple choices for each category. In this paper, we first study fast approximation algorithms for the trip planning query in a metric space, assuming that the data set fits in main memory, and give the theory analysis of their approximation bounds. Then, the trip planning query is examined for data sets that do not fit in main memory and must be stored on disk. For the disk-resident data, we consider two cases. In one case, we assume that the points are located in Euclidean space and indexed with an Rtree. In the other case, we consider the problem of points that lie on the edges of a spatial network (e.g. road network) and the distance between two points is defined using the shortest distance over the network. Finally, we give an experimental evaluation of the proposed algorithms using synthetic data sets generated on real road networks

HD-Index: Pushing the Scalability-Accuracy Boundary for Approximate kNN Search in High-Dimensional Spaces

Nearest neighbor searching of large databases in high-dimensional spaces is inherently difficult due to the curse of dimensionality. A flavor of approximation is, therefore, necessary to practically solve the problem of nearest neighbor search. In this paper, we propose a novel yet simple indexing scheme, HD-Index, to solve the problem of approximate k-nearest neighbor queries in massive high-dimensional databases. HD-Index consists of a set of novel hierarchical structures called RDB-trees built on Hilbert keys of database objects. The leaves of the RDB-trees store distances of database objects to reference objects, thereby allowing efficient pruning using distance filters. In addition to triangular inequality, we also use Ptolemaic inequality to produce better lower bounds. Experiments on massive (up to billion scale) high-dimensional (up to 1000+) datasets show that HD-Index is effective, efficient, and scalable.Comment: PVLDB 11(8):906-919, 201

Lower Bounds for Oblivious Near-Neighbor Search

We prove an $\Omega(d \lg n/ (\lg\lg n)^2)$ lower bound on the dynamic cell-probe complexity of statistically $\mathit{oblivious}$ approximate-near-neighbor search ($\mathsf{ANN}$) over the $d$-dimensional Hamming cube. For the natural setting of $d = \Theta(\log n)$, our result implies an $\tilde{\Omega}(\lg^2 n)$ lower bound, which is a quadratic improvement over the highest (non-oblivious) cell-probe lower bound for $\mathsf{ANN}$. This is the first super-logarithmic $\mathit{unconditional}$ lower bound for $\mathsf{ANN}$ against general (non black-box) data structures. We also show that any oblivious $\mathit{static}$ data structure for decomposable search problems (like $\mathsf{ANN}$) can be obliviously dynamized with $O(\log n)$ overhead in update and query time, strengthening a classic result of Bentley and Saxe (Algorithmica, 1980).Comment: 28 page

Complex queries and complex data

With the widespread availability of wearable computers, equipped with sensors such as GPS or cameras, and with the ubiquitous presence of micro-blogging platforms, social media sites and digital marketplaces, data can be collected and shared on a massive scale. A necessary building block for taking advantage from this vast amount of information are efficient and effective similarity search algorithms that are able to find objects in a database which are similar to a query object. Due to the general applicability of similarity search over different data types and applications, the formalization of this concept and the development of strategies for evaluating similarity queries has evolved to an important field of research in the database community, spatio-temporal database community, and others, such as information retrieval and computer vision. This thesis concentrates on a special instance of similarity queries, namely k-Nearest Neighbor (kNN) Queries and their close relative, Reverse k-Nearest Neighbor (RkNN) Queries. As a first contribution we provide an in-depth analysis of the RkNN join. While the problem of reverse nearest neighbor queries has received a vast amount of research interest, the problem of performing such queries in a bulk has not seen an in-depth analysis so far. We first formalize the RkNN join, identifying its monochromatic and bichromatic versions and their self-join variants. After pinpointing the monochromatic RkNN join as an important and interesting instance, we develop solutions for this class, including a self-pruning and a mutual pruning algorithm. We then evaluate these algorithms extensively on a variety of synthetic and real datasets. From this starting point of similarity queries on certain data we shift our focus to uncertain data, addressing nearest neighbor queries in uncertain spatio-temporal databases. Starting from the traditional definition of nearest neighbor queries and a data model for uncertain spatio-temporal data, we develop efficient query mechanisms that consider temporal dependencies during query evaluation. We define intuitive query semantics, aiming not only at returning the objects closest to the query but also their probability of being a nearest neighbor. After theoretically evaluating these query predicates we develop efficient querying algorithms for the proposed query predicates. Given the findings of this research on nearest neighbor queries, we extend these results to reverse nearest neighbor queries. Finally we address the problem of querying large datasets containing set-based objects, namely image databases, where images are represented by (multi-)sets of vectors and additional metadata describing the position of features in the image. We aim at reducing the number of kNN queries performed during query processing and evaluate a modified pipeline that aims at optimizing the query accuracy at a small number of kNN queries. Additionally, as feature representations in object recognition are moving more and more from the real-valued domain to the binary domain, we evaluate efficient indexing techniques for binary feature vectors.Nicht nur durch die Verbreitung von tragbaren Computern, die mit einer Vielzahl von Sensoren wie GPS oder Kameras ausgestattet sind, sondern auch durch die breite Nutzung von Microblogging-Plattformen, Social-Media Websites und digitale Marktplätze wie Amazon und Ebay wird durch die User eine gigantische Menge an Daten veröffentlicht. Um aus diesen Daten einen Mehrwert erzeugen zu können bedarf es effizienter und effektiver Algorithmen zur Ähnlichkeitssuche, die zu einem gegebenen Anfrageobjekt ähnliche Objekte in einer Datenbank identifiziert. Durch die Allgemeinheit dieses Konzeptes der Ähnlichkeit über unterschiedliche Datentypen und Anwendungen hinweg hat sich die Ähnlichkeitssuche zu einem wichtigen Forschungsfeld, nicht nur im Datenbankumfeld oder im Bereich raum-zeitlicher Datenbanken, sondern auch in anderen Forschungsgebieten wie dem Information Retrieval oder dem Maschinellen Sehen entwickelt. In der vorliegenden Arbeit beschäftigen wir uns mit einem speziellen Anfrageprädikat im Bereich der Ähnlichkeitsanfragen, mit k-nächste Nachbarn (kNN) Anfragen und ihrem Verwandten, den Revers k-nächsten Nachbarn (RkNN) Anfragen. In einem ersten Beitrag analysieren wir den RkNN Join. Obwohl das Problem von reverse nächsten Nachbar Anfragen in den letzten Jahren eine breite Aufmerksamkeit in der Forschungsgemeinschaft erfahren hat, wurde das Problem eine Menge von RkNN Anfragen gleichzeitig auszuführen nicht ausreichend analysiert. Aus diesem Grund formalisieren wir das Problem des RkNN Joins mit seinen monochromatischen und bichromatischen Varianten. Wir identifizieren den monochromatischen RkNN Join als einen wichtigen und interessanten Fall und entwickeln entsprechende Anfragealgorithmen. In einer detaillierten Evaluation vergleichen wir die ausgearbeiteten Verfahren auf einer Vielzahl von synthetischen und realen Datensätzen. Nach diesem Kapitel über Ähnlichkeitssuche auf sicheren Daten konzentrieren wir uns auf unsichere Daten, speziell im Bereich raum-zeitlicher Datenbanken. Ausgehend von der traditionellen Definition von Nachbarschaftsanfragen und einem Datenmodell für unsichere raum-zeitliche Daten entwickeln wir effiziente Anfrageverfahren, die zeitliche Abhängigkeiten bei der Anfragebearbeitung beachten. Zu diesem Zweck definieren wir Anfrageprädikate die nicht nur die Objekte zurückzugeben, die dem Anfrageobjekt am nächsten sind, sondern auch die Wahrscheinlichkeit mit der sie ein nächster Nachbar sind. Wir evaluieren die definierten Anfrageprädikate theoretisch und entwickeln effiziente Anfragestrategien, die eine Anfragebearbeitung zu vertretbaren Laufzeiten gewährleisten. Ausgehend von den Ergebnissen für Nachbarschaftsanfragen erweitern wir unsere Ergebnisse auf Reverse Nachbarschaftsanfragen. Zuletzt behandeln wir das Problem der Anfragebearbeitung bei Mengen-basierten Objekten, die zum Beispiel in Bilddatenbanken Verwendung finden: Oft werden Bilder durch eine Menge von Merkmalsvektoren und zusätzliche Metadaten (zum Beispiel die Position der Merkmale im Bild) dargestellt. Wir evaluieren eine modifizierte Pipeline, die darauf abzielt, die Anfragegenauigkeit bei einer kleinen Anzahl an kNN-Anfragen zu maximieren. Da reellwertige Merkmalsvektoren im Bereich der Objekterkennung immer öfter durch Bitvektoren ersetzt werden, die sich durch einen geringeren Speicherplatzbedarf und höhere Laufzeiteffizienz auszeichnen, evaluieren wir außerdem Indexierungsverfahren für Binärvektoren