The Transitive Minimum Manhattan Subnetwork Problem in 3 Dimensions

We consider the Minimum Manhattan Subnetwork (MMSN) Problem which generalizes the already known Minimum Manhattan Network (MMN) Problem: Given a set P of n points in the plane, find shortest rectilinear paths between all pairs of points. These paths form a network, the total length of which has to be minimized. From a graph theoretical point of view, a MMN is a 1-spanner with respect to the L_1 metric. In contrast to the MMN problem, a solution to the MMSN problem does not demand L_1 -shortest paths for all point pairs, but only for a given set R subseteq P imes P of pairs. The complexity status of the MMN problem is still unsolved in geq 2 dimensions, whereas the MMSN was shown to be NP -complete considering general relations R in the plane. We restrict the MMSN problem to transitive relations R_T ({em Transitive} Minimum Manhattan Subnetwork (TMMSN) Problem) and show that the TMMSN problem is Max-SNP -complete with epsilon<frac{1}{8} in 3 dimensions

The Transitive Minimum Manhattan Subnetwork Problem in 3 Dimensions

We consider the Minimum Manhattan Subnetwork (MMSN) Problem which generalizes the already known Minimum Manhattan Network (MMN) Problem: Given a set P of n points in the plane, find shortest rectilinear paths between all pairs of points. These paths form a network, the total length of which has to be minimized. From a graph theoretical point of view, a MMN is a 1-spanner with respect to the L_1 metric. In contrast to the MMN problem, a solution to the MMSN problem does not demand L_1 -shortest paths for all point pairs, but only for a given set R subseteq P imes P of pairs. The complexity status of the MMN problem is still unsolved in geq 2 dimensions, whereas the MMSN was shown to be NP -complete considering general relations R in the plane. We restrict the MMSN problem to transitive relations R_T ({em Transitive} Minimum Manhattan Subnetwork (TMMSN) Problem) and show that the TMMSN problem is Max-SNP -complete with epsilon<frac{1}{8} in 3 dimensions

Algorithm engineering in geometric network planning and data mining

The geometric nature of computational problems provides a rich source of solution strategies as well as complicating obstacles. This thesis considers three problems in the context of geometric network planning, data mining and spherical geometry. Geometric Network Planning: In the d-dimensional Generalized Minimum Manhattan Network problem (d-GMMN) one is interested in finding a minimum cost rectilinear network N connecting a given set of n pairs of points in ℝ^d such that each pair is connected in N via a shortest Manhattan path. The decision version of this optimization problem is known to be NP-hard. The best known upper bound is an O(log^{d+1} n) approximation for d>2 and an O(log n) approximation for 2-GMMN. In this work we provide some more insight in, whether the problem admits constant factor approximations in polynomial time. We develop two new algorithms, a `scale-diversity aware' algorithm with an O(D) approximation guarantee for 2-GMMN. Here D is a measure for the different `scales' that appear in the input, D ∈ O(log n) but potentially much smaller, depending on the problem instance. The other algorithm is based on a primal-dual scheme solving a more general, combinatorial problem - which we call Path Cover. On 2-GMMN it performs well in practice with good a posteriori, instance-based approximation guarantees. Furthermore, it can be extended to deal with obstacle avoiding requirements. We show that the Path Cover problem is at least as hard to approximate as the Hitting Set problem. Moreover, we show that solutions of the primal-dual algorithm are 4ω^2 approximations, where ω ≤ n denotes the maximum overlap of a problem instance. This implies that a potential proof of O(1)-inapproximability for 2-GMMN requires gadgets of many different scales and non-constant overlap in the construction. Geometric Map Matching for Heterogeneous Data: For a given sequence of location measurements, the goal of the geometric map matching is to compute a sequence of movements along edges of a spatially embedded graph which provides a `good explanation' for the measurements. The problem gets challenging as real world data, like traces or graphs from the OpenStreetMap project, does not exhibit homogeneous data quality. Graph details and errors vary in areas and each trace has changing noise and precision. Hence, formalizing what a `good explanation' is becomes quite difficult. We propose a novel map matching approach, which locally adapts to the data quality by constructing what we call dominance decompositions. While our approach is computationally more expensive than previous approaches, our experiments show that it allows for high quality map matching, even in presence of highly variable data quality without parameter tuning. Rational Points on the Unit Spheres: Each non-zero point in ℝ^d identifies a closest point x on the unit sphere S^{d-1}. We are interested in computing an ε-approximation y ∈ ℚ^d for x, that is exactly on S^{d-1} and has low bit-size. We revise lower bounds on rational approximations and provide explicit spherical instances. We prove that floating-point numbers can only provide trivial solutions to the sphere equation in ℝ^2 and ℝ^3. However, we show how to construct a rational point with denominators of at most 10(d-1)/ε^2 for any given ε ∈ (0, 1/8], improving on a previous result. The method further benefits from algorithms for simultaneous Diophantine approximation. Our open-source implementation and experiments demonstrate the practicality of our approach in the context of massive data sets, geo-referenced by latitude and longitude values.Die geometrische Gestalt von Berechnungsproblemen liefert vielfältige Lösungsstrategieen aber auch Hindernisse. Diese Arbeit betrachtet drei Probleme im Gebiet der geometrischen Netzwerk Planung, des geometrischen Data Minings und der sphärischen Geometrie. Geometrische Netzwerk Planung: Im d-dimensionalen Generalized Minimum Manhattan Network Problem (d-GMMN) möchte man ein günstigstes geradliniges Netzwerk finden, welches jedes der gegebenen n Punktepaare aus ℝ^d mit einem kürzesten Manhattan Pfad verbindet. Es ist bekannt, dass die Entscheidungsvariante dieses Optimierungsproblems NP-hart ist. Die beste bekannte obere Schranke ist eine O(log^{d+1} n) Approximation für d>2 und eine O(log n) Approximation für 2-GMMN. Durch diese Arbeit geben wir etwas mehr Einblick, ob das Problem eine Approximation mit konstantem Faktor in polynomieller Zeit zulässt. Wir entwickeln zwei neue Algorithmen. Ersterer nutzt die `Skalendiversität' und hat eine O(D) Approximationsgüte für 2-GMMN. Hierbei ist D ein Maß für die in Eingaben auftretende `Skalen'. D ∈ O(log n), aber potentiell deutlichen kleiner für manche Problem Instanzen. Der andere Algorithmus basiert auf einem Primal-Dual Schema zur Lösung eines allgemeineren, kombinatorischen Problems, welches wir Path Cover nennen. Die praktisch erzielten a posteriori Approximationsgüten auf Instanzen von 2-GMMN verhalten sich gut. Dieser Algorithmus kann für Netzwerk Planungsprobleme mit Hindernis-Anforderungen angepasst werden. Wir zeigen, dass das Path Cover Problem mindestens so schwierig zu approximieren ist wie das Hitting Set Problem. Darüber hinaus zeigen wir, dass Lösungen des Primal-Dual Algorithmus 4ω^2 Approximationen sind, wobei ω ≤ n die maximale Überlappung einer Probleminstanz bezeichnet. Daher müssen potentielle Beweise, die konstante Approximationen für 2-GMMN ausschließen möchten, Instanzen mit vielen unterschiedlichen Skalen und nicht konstanter Überlappung konstruieren. Geometrisches Map Matching für heterogene Daten: Für eine gegebene Sequenz von Positionsmessungen ist das Ziel des geometrischen Map Matchings eine Sequenz von Bewegungen entlang Kanten eines räumlich eingebetteten Graphen zu finden, welche eine `gute Erklärung' für die Messungen ist. Das Problem wird anspruchsvoll da reale Messungen, wie beispielsweise Traces oder Graphen des OpenStreetMap Projekts, keine homogene Datenqualität aufweisen. Graphdetails und -fehler variieren in Gebieten und jeder Trace hat wechselndes Rauschen und Messgenauigkeiten. Zu formalisieren, was eine `gute Erklärung' ist, wird dadurch schwer. Wir stellen einen neuen Map Matching Ansatz vor, welcher sich lokal der Datenqualität anpasst indem er sogenannte Dominance Decompositions berechnet. Obwohl unser Ansatz teurer im Rechenaufwand ist, zeigen unsere Experimente, dass qualitativ hochwertige Map Matching Ergebnisse auf hoch variabler Datenqualität erzielbar sind ohne vorher Parameter kalibrieren zu müssen. Rationale Punkte auf Einheitssphären: Jeder, von Null verschiedene, Punkt in ℝ^d identifiziert einen nächsten Punkt x auf der Einheitssphäre S^{d-1}. Wir suchen eine ε-Approximation y ∈ ℚ^d für x zu berechnen, welche exakt auf S^{d-1} ist und niedrige Bit-Größe hat. Wir wiederholen untere Schranken an rationale Approximationen und liefern explizite, sphärische Instanzen. Wir beweisen, dass Floating-Point Zahlen nur triviale Lösungen zur Sphären-Gleichung in ℝ^2 und ℝ^3 liefern können. Jedoch zeigen wir die Konstruktion eines rationalen Punktes mit Nennern die maximal 10(d-1)/ε^2 sind für gegebene ε ∈ (0, 1/8], was ein bekanntes Resultat verbessert. Darüber hinaus profitiert die Methode von Algorithmen für simultane Diophantische Approximationen. Unsere quell-offene Implementierung und die Experimente demonstrieren die Praktikabilität unseres Ansatzes für sehr große, durch geometrische Längen- und Breitengrade referenzierte, Datensätze

Multi-level characterization and information extraction in directed and node-labeled functional brain networks

Current research in computational neuroscience puts great emphasis on the computation and analysis of the functional connectivity of the brain. The methodological developments presented in this work are concerned with a group-specific comprehensive analysis of networks that represent functional interaction patterns. Four application studies are presented, in which functional brain network samples of different clinical background were analyzed in different ways, using combinations of established approaches and own methodological developments. Study I is concerned with a sample-specific decomposition of the functional brain networks of depressed subjects and healthy controls into small functionally important and recurring subnetworks (motifs) using own developments. Study II investigates whether lithium treatment effects are reflected in the functional brain networks of HIV-positive subjects with diagnosed cognitive impairment. For it, microscopic and macroscopic structural properties were analyzed. Study III explores spatially highly resolved functional brain networks with regard to a functional segmentation given by identified module (community) structure. Also, ground truth networks with known module structure were generated using own methodological developments. They formed the basis of a comprehensive simulation study that quantified module structure quality and preservation in order to evaluate the effects of a novel approach for the identification of connectivity (lsGCI). Study IV tracks the time-evolution of module structure and introduces a newly developed own approach for the determination of edge weight thresholds based on multicriteria optimization. The methodological challenges that underly these different topological analyses, but also the various opportunities to gain an improved understanding of neural information processing among brain areas were highlighted by this work and the presented results

Automated circuit extraction from mask descriptions of MOS networks

Also issued as Thesis (M.S.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1984.Includes bibliographical references (p. 122-123).Supported in part by U.S. Air Force, Office of Scientific Research. F49620-81-C-0054 F49620-84-C-0004. Supported in part by the Bell Labs Fellowship in Applied Computer Science.Steven Paul McCormick

Visualization of Metabolic Networks

The metabolism constitutes the universe of biochemical reactions taking place in a cell of an organism. These processes include the synthesis, transformation, and degradation of molecules for an organism to grow, to reproduce and to interact with its environment. A good way to capture the complexity of these processes is the representation as metabolic network, in which sets of molecules are transformed into products by a chemical reaction, and the products are being processed further. The underlying graph model allows a structural analysis of this network using established graphtheoretical algorithms on the one hand, and a visual representation by applying layout algorithms combined with information visualization techniques on the other. In this thesis we will take a look at three different aspects of graph visualization within the context of biochemical systems: the representation and interactive exploration of static networks, the visual analysis of dynamic networks, and the comparison of two network graphs. We will demonstrate, how established infovis techniques can be combined with new algorithms and applied to specific problems in the area of metabolic network visualization. We reconstruct the metabolic network covering the complete set of chemical reactions present in a generalized eucaryotic cell from real world data available from a popular metabolic pathway data base and present a suitable data structure. As the constructed network is very large, it is not feasible for the display as a whole. Instead, we introduce a technique to analyse this static network in a top-down approach starting with an overview and displaying detailed reaction networks on demand. This exploration method is also applied to compare metabolic networks in different species and from different resources. As for the analysis of dynamic networks, we present a framework to capture changes in the connectivity as well as changes in the attributes associated with the network’s elements