Tight(er) bounds for similarity measures, smoothed approximation and broadcasting

In this thesis, we prove upper and lower bounds on the complexity of sequence similarity measures, the approximability of geometric problems on realistic inputs, and the performance of randomized broadcasting protocols. The first part approaches the question why a number of fundamental polynomial-time problems - specifically, Dynamic Time Warping, Longest Common Subsequence (LCS), and the Levenshtein distance - resists decades-long attempts to obtain polynomial improvements over their simple dynamic programming solutions. We prove that any (strongly) subquadratic algorithm for these and related sequence similarity measures would refute the Strong Exponential Time Hypothesis (SETH). Focusing particularly on LCS, we determine a tight running time bound (up to lower order factors and conditional on SETH) when the running time is expressed in terms of all input parameters that have been previously exploited in the extensive literature. In the second part, we investigate the approximation performance of the popular 2-Opt heuristic for the Traveling Salesperson Problem using the smoothed analysis paradigm. For the Fréchet distance, we design an improved approximation algorithm for the natural input class of c-packed curves, matching a conditional lower bound. Finally, in the third part we prove tighter performance bounds for processes that disseminate a piece of information, either as quickly as possible (rumor spreading) or as anonymously as possible (cryptogenography).Die vorliegende Dissertation beweist obere und untere Schranken an die Komplexität von Sequenzähnlichkeitsmaßen, an die Approximierbarkeit geometrischer Probleme auf realistischen Eingaben und an die Effektivität randomisierter Kommunikationsprotokolle. Der erste Teil befasst sich mit der Frage, warum für eine Vielzahl fundamentaler Probleme im Polynomialzeitbereich - insbesondere für das Dynamic-Time-Warping, die längste gemeinsame Teilfolge (LCS) und die Levenshtein-Distanz - seit Jahrzehnten keine Algorithmen gefunden werden konnten, die polynomiell schneller sind als ihre einfachen Lösungen mittels dynamischer Programmierung. Wir zeigen, dass ein (im strengen Sinne) subquadratischer Algorithmus für diese und verwandte Ähnlichkeitsmaße die starke Exponentialzeithypothese (SETH) widerlegen würde. Für LCS zeigen wir eine scharfe Schranke an die optimale Laufzeit (unter der SETH und bis auf Faktoren niedrigerer Ordnung) in Abhängigkeit aller bisher untersuchten Eingabeparameter. Im zweiten Teil untersuchen wir die Approximationsgüte der klassischen 2-Opt-Heuristik für das Problem des Handlungsreisenden anhand des Smoothed-Analysis-Paradigmas. Weiterhin entwickeln wir einen verbesserten Approximationsalgorithmus für die Fréchet-Distanz auf einer Klasse natürlicher Eingaben. Der letzte Teil beweist neue Schranken für die Effektivität von Prozessen, die Informationen entweder so schnell wie möglich (Rumor-Spreading) oder so anonym wie möglich (Kryptogenografie) verbreiten

Improving grid based quantum dynamics

In this work the development and refinement of methods that allow for a more efficient use of quantum dynamics calculations with the dynamic Fourier method (DFM) and the extension of the DFMs applicability to solvated systems are presented. The systems studied with these methods are primarily molecular reactions. The contents of this thesis can be divided in three parts. In the first, approaches to automate the necessary and difficult construction of reactive coordinates are developed. The reactive coordinates constructed are linear combinations of the Cartesian coordinates of the atoms in two cases and nonlinear combinations in one case. Of the two linear approaches, one uses points along the minimum energy path of the reaction to span a reactive subspace and the other uses classical trajectories that are run along the reaction path to extract essential motions. The nonlinear approach uses an autoencoder that learns an efficient low-dimensional description of the reactive space by using large amounts of trajectory data. All three methods are presented in detail and applied to example systems. The advantages of each method and the immense potential of the nonlinear approach are discussed. In the second part, the inclusion of dynamic solvent effects on the reactive solute is studied. This is important, because dynamic interactions can significantly alter the outcome of reactions. Three methods were developed in the course of this work. The first one treats the solvent implicitly as a continuum that causes frictional forces acting on the solute. This is computationally convenient but, as it cannot describe certain interactions -- such as collisions -- it is not suited for all systems. The second and third method treat the solvent particles explicitly, using frozen and classically propagated environments, respectively. Here, the third approach extends the second to a much larger field of application by means of a quantum-classical TDSCF. The three methods are applied to the practically important problems of the photogeneration of diphenylmethyl cations as reactive intermediates and the photorelaxation of uracil as a way to prevent photodamage in RNA. The third part is comprised of two minor improvements. The first deals with errors that can be introduced due to an incorrect treatment of an approximation within the Wilson G-matrix formalism, a formalism that offers a simple way to perform coordinate transformations. This not only applies to some implementations of the DFM method, but to all methods that use the G-matrix formalism with nonlinear coordinates. The second studies the use of undersampling in the DFM method to reduce the number of grid points and thus computational time. Its potential savings are demonstrated using a model system