FPT-Algorithms for Computing Gromov-Hausdorff and Interleaving Distances Between Trees

The Gromov-Hausdorff distance is a natural way to measure the distortion between two metric spaces. However, there has been only limited algorithmic development to compute or approximate this distance. We focus on computing the Gromov-Hausdorff distance between two metric trees. Roughly speaking, a metric tree is a metric space that can be realized by the shortest path metric on a tree. Any finite tree with positive edge weight can be viewed as a metric tree where the weight is treated as edge length and the metric is the induced shortest path metric in the tree. Previously, Agarwal et al. showed that even for trees with unit edge length, it is NP-hard to approximate the Gromov-Hausdorff distance between them within a factor of 3. In this paper, we present a fixed-parameter tractable (FPT) algorithm that can approximate the Gromov-Hausdorff distance between two general metric trees within a multiplicative factor of 14. Interestingly, the development of our algorithm is made possible by a connection between the Gromov-Hausdorff distance for metric trees and the interleaving distance for the so-called merge trees. The merge trees arise in practice naturally as a simple yet meaningful topological summary (it is a variant of the Reeb graphs and contour trees), and are of independent interest. It turns out that an exact or approximation algorithm for the interleaving distance leads to an approximation algorithm for the Gromov-Hausdorff distance. One of the key contributions of our work is that we re-define the interleaving distance in a way that makes it easier to develop dynamic programming approaches to compute it. We then present a fixed-parameter tractable algorithm to compute the interleaving distance between two merge trees exactly, which ultimately leads to an FPT-algorithm to approximate the Gromov-Hausdorff distance between two metric trees. This exact FPT-algorithm to compute the interleaving distance between merge trees is of interest itself, as it is known that it is NP-hard to approximate it within a factor of 3, and previously the best known algorithm has an approximation factor of O(sqrt{n}) even for trees with unit edge length

New and Improved Algorithms for Unordered Tree Inclusion

The tree inclusion problem is, given two node-labeled trees P and T (the "pattern tree" and the "text tree"), to locate every minimal subtree in T (if any) that can be obtained by applying a sequence of node insertion operations to P. Although the ordered tree inclusion problem is solvable in polynomial time, the unordered tree inclusion problem is NP-hard. The currently fastest algorithm for the latter is from 1995 and runs in O(poly(m,n) * 2^{2d}) = O^*(2^{2d}) time, where m and n are the sizes of the pattern and text trees, respectively, and d is the maximum outdegree of the pattern tree. Here, we develop a new algorithm that improves the exponent 2d to d by considering a particular type of ancestor-descendant relationships and applying dynamic programming, thus reducing the time complexity to O^*(2^d). We then study restricted variants of the unordered tree inclusion problem where the number of occurrences of different node labels and/or the input trees\u27 heights are bounded. We show that although the problem remains NP-hard in many such cases, it can be solved in polynomial time for c = 2 and in O^*(1.8^d) time for c = 3 if the leaves of P are distinctly labeled and each label occurs at most c times in T. We also present a randomized O^*(1.883^d)-time algorithm for the case that the heights of P and T are one and two, respectively

Efficient Similarity Search in Structured Data

Modern database applications are characterized by two major aspects: the use of complex data types with internal structure and the need for new data analysis methods. The focus of database users has shifted from simple queries to complex analyses of the data, known as knowledge discovery in databases. Important tasks in this area are the grouping of data objects (clustering), the classification of new data objects or the detection of exceptional data objects (outlier detection). Most algorithms for solving those problems are based on similarity search in databases. This makes efficient similarity search in large databases of structured objects an important basic operation for modern database applications. In this thesis we develop efficient methods for similarity search in large databases of structured data and improve the efficiency of existing query processing techniques. For the data objects, only a tree or graph structure is assumed which can be extended with arbitrary attribute information. Starting with an analysis of the demands from two example applications, several important requirements for similarity measures are identified. One aspect is the adaptability of the similarity search method to the requirements of the user and the application domain. This can even imply a change of the similarity measure between two successive queries of the same user. An explanation component which makes clear why objects are considered similar by the system is a necessary precondition for a purposeful adaption of the measure. Consequently, the edit distance, well-known from string processing, is a common similarity measure for graph structured objects. Its feature to allow a visualization of corresponding substructures and the possibility to weight single operations are the reason for this popularity. But it turns out that the edit distance and similar measures for tree structures are computationally extremely complex which makes them unsuitable for today's large and even growing databases. Therefore, we develop a multi-step query processing architecture which reduces the number of necessary distance calculations significantly. This is achieved by employing suitable filter methods. Furthermore, we show that by easing certain restrictions on the similarity measure, a significant performance gain can be obtained without reducing the quality of the measure. To achieve this, matchings of substructures (vertices or edges) of the data objects are determined. An additional cost function for those matchings allows to derive a similarity measure for structured data, called the edge matching distance, from the cost optimal matching of the substructures. But even for this new similarity measure, efficiency can be improved significantly by using a multi-step query processing approach. This allows the use of the edge matching distance for knowledge discovery applications in large databases. Within the thesis, the properties of our new similarity search methods are proved both theoretically and through experiments.Moderne Datenbankanwendungen werden vor allem durch zwei wesentliche Aspekte charakterisiert. Dies ist zum einen die Verwendung komplexer Datentypen mit interner Struktur und zum anderen die Notwendigkeit neuer Recherchemöglichkeiten. Der Fokus bei der Datenbankbenutzung hat sich von einfachen Anfragen hin zu komplexen Analysen des Datenbestandes, dem sogenannten Knowledge-Discovery in Datenbanken, entwickelt. Wichtige Analysetechniken in diesem Bereich sind unter anderem die Gruppierung der Daten in Teilmengen (Clustering), die Klassifikation neuer Datenobjekte im Bezug auf den vorhandenen Datenbestand und das Erkennen von Ausreißern in den Daten (Outlier-Identifikation). Die Basis für die meisten Verfahren zur Lösung dieser Aufgaben bildet dabei die Bestimmung der Ähnlichkeit von Datenbankobjekten. Die effiziente Ähnlichkeitssuche in großen Datenbanken strukturierter Objekte ist daher eine wichtige Basisoperation für moderne Datenbankanwendungen. In dieser Doktorarbeit werden daher effiziente Verfahren für die Ähnlichkeitssuche in großen Mengen strukturierter Objekte entwickelt, bzw. die Effizienz vorhandener Verfahren deutlich zu verbessert. Dabei wird lediglich eine baum- oder allgemein graphartige innere Struktur der Datenobjekte vorausgesetzt, die durch beliebige Attribute erweitert wird. Ausgehend von einer Analyse der Anforderungen an Ähnlichkeitssuchverfahren in zwei Beispielsanwendungen aus dem Bereich der Bildsuche und des Proteindockings, wurden mehrere wichtige Aspekte der Ähnlichkeitssuche identifiziert. Ein erster Aspekt ist, das Maß für die Ähnlichkeit für den Benutzer anpassbar zu gestalten, da der zugrundeliegende Ähnlichkeitsbegriff sowohl benutzer- als auch situationsabhängig ist, was bis hin zur Änderung des Ähnlichkeitsbegriffs zwischen zwei aufeinanderfolgenden Anfragen gehen kann. Voraussetzung für eine zielgerichtete Anpassung des Ähnlichkeitsbegriffs ist dabei eine Erklärungskomponente, welche dem Benutzer das Zustandekommen eines Ähnlichkeitswertes verdeutlicht. Die aus der Stringverarbeitung bekannte Edit-Distanz ist deshalb ein weit verbreitetes Maß für die Ähnlichkeit von graphstrukturierten Objekten, da sie eine Gewichtung einzelner Operationen erlaubt und durch eine Zuordnung von Teilobjekten aus den zu vergleichenden Strukturen eine Erklärungskomponente liefert. Es zeigt sich jedoch, dass die Bestimmung der Edit-Distanz und vergleichbarer Ähnlichkeitsmaße für Baum- oder Graphstrukturen extrem zeitaufwendig ist. Es wird daher zunächst ein mehrstufiges Anfragebearbeitungsmodell entwickelt, welches durch geeignete Filterschritte die Anzahl der notwendigen Distanzberechnungen massiv reduziert und so die Geschwindigkeit der Anfragebearbeitung deutlich steigert bzw. erst für große Datenmengen akzeptabel macht. Im nächsten Schritt wird aufgezeigt, wie sich durch Lockerung einiger Bedingungen für das Ähnlichkeitsmaß deutliche Geschwindigkeitssteigerungen erreichen lassen, ohne Einbußen bezüglich der Qualität der Anfrageergebnisse hinnehmen zu müssen. Dazu werden Paarungen von Teilstrukturen (Knoten oder Kanten) der zu vergleichenden Objekte bestimmt, die zusätzlich mittels einer Kostenfunktion gewichtet werden. Eine bezüglich dieser Kostenfunktion optimale Paarung aller Teilstrukturen stellt dann ein Maß für die Ähnlichkeit der Vergleichsobjekte dar, die sogenannte "edge matching distance". Es zeigt sich jedoch, dass auch für dieses neue Ähnlichkeitsmaß eine mehrstufige Anfragebearbeitung zusammen mit entsprechenden, neuartigen Filtermethoden eine erhebliche Performanzsteigerung erlaubt. Diese stellt die Voraussetzung für die Anwendung der Verfahren im Rahmen des Knowledge-Discovery in großen Datenbanken dar. Dabei werden die genannten Eigenschaften der neu entwickelten Verfahren sowohl theoretisch als auch mittels praktischer Experimente belegt

Fine-grained Complexity Meets IP = PSPACE

In this paper we study the fine-grained complexity of finding exact and approximate solutions to problems in P. Our main contribution is showing reductions from exact to approximate solution for a host of such problems. As one (notable) example, we show that the Closest-LCS-Pair problem (Given two sets of strings $A$ and $B$, compute exactly the maximum $\textsf{LCS}(a, b)$ with $(a, b) \in A \times B$) is equivalent to its approximation version (under near-linear time reductions, and with a constant approximation factor). More generally, we identify a class of problems, which we call BP-Pair-Class, comprising both exact and approximate solutions, and show that they are all equivalent under near-linear time reductions. Exploring this class and its properties, we also show: $\bullet$ Under the NC-SETH assumption (a significantly more relaxed assumption than SETH), solving any of the problems in this class requires essentially quadratic time. $\bullet$ Modest improvements on the running time of known algorithms (shaving log factors) would imply that NEXP is not in non-uniform $\textsf{NC}^1$. $\bullet$ Finally, we leverage our techniques to show new barriers for deterministic approximation algorithms for LCS. At the heart of these new results is a deep connection between interactive proof systems for bounded-space computations and the fine-grained complexity of exact and approximate solutions to problems in P. In particular, our results build on the proof techniques from the classical IP = PSPACE result

Online Analysis of Dynamic Streaming Data

Die Arbeit zum Thema "Online Analysis of Dynamic Streaming Data" beschäftigt sich mit der Distanzmessung dynamischer, semistrukturierter Daten in kontinuierlichen Datenströmen um Analysen auf diesen Datenstrukturen bereits zur Laufzeit zu ermöglichen. Hierzu wird eine Formalisierung zur Distanzberechnung für statische und dynamische Bäume eingeführt und durch eine explizite Betrachtung der Dynamik von Attributen einzelner Knoten der Bäume ergänzt. Die Echtzeitanalyse basierend auf der Distanzmessung wird durch ein dichte-basiertes Clustering ergänzt, um eine Anwendung des Clustering, einer Klassifikation, aber auch einer Anomalieerkennung zu demonstrieren. Die Ergebnisse dieser Arbeit basieren auf einer theoretischen Analyse der eingeführten Formalisierung von Distanzmessungen für dynamische Bäume. Diese Analysen werden unterlegt mit empirischen Messungen auf Basis von Monitoring-Daten von Batchjobs aus dem Batchsystem des GridKa Daten- und Rechenzentrums. Die Evaluation der vorgeschlagenen Formalisierung sowie der darauf aufbauenden Echtzeitanalysemethoden zeigen die Effizienz und Skalierbarkeit des Verfahrens. Zudem wird gezeigt, dass die Betrachtung von Attributen und Attribut-Statistiken von besonderer Bedeutung für die Qualität der Ergebnisse von Analysen dynamischer, semistrukturierter Daten ist. Außerdem zeigt die Evaluation, dass die Qualität der Ergebnisse durch eine unabhängige Kombination mehrerer Distanzen weiter verbessert werden kann. Insbesondere wird durch die Ergebnisse dieser Arbeit die Analyse sich über die Zeit verändernder Daten ermöglicht

Efficient similarity computations on parallel machines using data shaping

Similarity computation is a fundamental operation in all forms of data. Big Data is, typically, characterized by attributes such as volume, velocity, variety, veracity, etc. In general, Big Data variety appears as structured, semi-structured or unstructured forms. The volume of Big Data in general, and semi-structured data in particular, is increasing at a phenomenal rate. Big Data phenomenon is posing new set of challenges to similarity computation problems occurring in semi-structured data. Technology and processor architecture trends suggest very strongly that future processors shall have ten\u27s of thousands of cores (hardware threads). Another crucial trend is that ratio between on-chip and off-chip memory to core counts is decreasing. State-of-the-art parallel computing platforms such as General Purpose Graphics Processors (GPUs) and MICs are promising for high performance as well high throughput computing. However, processing semi-structured component of Big Data efficiently using parallel computing systems (e.g. GPUs) is challenging. Reason being most of the emerging platforms (e.g. GPUs) are organized as Single Instruction Multiple Thread/Data machines which are highly structured, where several cores (streaming processors) operate in lock-step manner, or they require a high degree of task-level parallelism. We argue that effective and efficient solutions to key similarity computation problems need to operate in a synergistic manner with the underlying computing hardware. Moreover, semi-structured form input data needs to be shaped or reorganized with the goal to exploit the enormous computing power of \textit{state-of-the-art} highly threaded architectures such as GPUs. For example, shaping input data (via encoding) with minimal data-dependence can facilitate flexible and concurrent computations on high throughput accelerators/co-processors such as GPU, MIC, etc. We consider various instances of traditional and futuristic problems occurring in intersection of semi-structured data and data analytics. Preprocessing is an operation common at initial stages of data processing pipelines. Typically, the preprocessing involves operations such as data extraction, data selection, etc. In context of semi-structured data, twig filtering is used in identifying (and extracting) data of interest. Duplicate detection and record linkage operations are useful in preprocessing tasks such as data cleaning, data fusion, and also useful in data mining, etc., in order to find similar tree objects. Likewise, tree edit is a fundamental metric used in context of tree problems; and similarity computation between trees another key problem in context of Big Data. This dissertation makes a case for platform-centric data shaping as a potent mechanism to tackle the data- and architecture-borne issues in context of semi-structured data processing on GPU and GPU-like parallel architecture machines. In this dissertation, we propose several data shaping techniques for tree matching problems occurring in semi-structured data. We experiment with real world datasets. The experimental results obtained reveal that the proposed platform-centric data shaping approach is effective for computing similarities between tree objects using GPGPUs. The techniques proposed result in performance gains up to three orders of magnitude, subject to problem and platform

ExtrudeNet: Unsupervised Inverse Sketch-and-Extrude for Shape Parsing

Sketch-and-extrude is a common and intuitive modeling process in computer aided design. This paper studies the problem of learning the shape given in the form of point clouds by inverse sketch-and-extrude. We present ExtrudeNet, an unsupervised end-to-end network for discovering sketch and extrude from point clouds. Behind ExtrudeNet are two new technical components: 1) an effective representation for sketch and extrude, which can model extrusion with freeform sketches and conventional cylinder and box primitives as well; and 2) a numerical method for computing the signed distance field which is used in the network learning. This is the first attempt that uses machine learning to reverse engineer the sketch-and-extrude modeling process of a shape in an unsupervised fashion. ExtrudeNet not only outputs a compact, editable and interpretable representation of the shape that can be seamlessly integrated into modern CAD software, but also aligns with the standard CAD modeling process facilitating various editing applications, which distinguishes our work from existing shape parsing research. Code is released at https://github.com/kimren227/ExtrudeNet.Comment: Accepted to ECCV 202

Efficient Change Management of XML Documents

XML-based documents play a major role in modern information architectures and their corresponding work-flows. In this context, the ability to identify and represent differences between two versions of a document is essential. A second important aspect is the merging of document versions, which becomes crucial in parallel editing processes. Many different approaches exist that meet these challenges. Most rely on operational transformation or document annotation. In both approaches, the operations leading to changes are tracked, which requires corresponding editing applications. In the context of software development, however, a state-based approach is common. Here, document versions are compared and merged using external tools, called diff and patch. This allows users for freely editing documents without being tightened to special tools. Approaches exist that are able to compare XML documents. A corresponding merge capability is still not available. In this thesis, I present a comprehensive framework that allows for comparing and merging of XML documents using a state-based approach. Its design is based on an analysis of XML documents and their modification patterns. The heart of the framework is a context-oriented delta model. I present a diff algorithm that appears to be highly efficient in terms of speed and delta quality. The patch algorithm is able to merge document versions efficiently and reliably. The efficiency and the reliability of my approach are verified using a competitive test scenario

Enhancement of Query processing on XML data

Ph.DDOCTOR OF PHILOSOPH