We study a generalization of the problem of finding bounds on the number of discrete chains, which itself is a generalization of the Erdős unit distance problem. Given a set of points in Euclidean space and a tree graph consisting of a much smaller number of vertices, we study the maximum possible number of tree graphs which can be represented by a prescribed tree graph. We derive an algorithm for finding tight bounds for this family of problems up to chain bound discrepancy, and give upper and lower bounds in special cases. / Master of Science / We study a generalization of the problem of finding bounds on the number of discrete chains, which itself is a generalization of the Erdős unit distance problem, a famous mathematics problem named after mathematician Paul Erdős. Given a set of points, and a tree graph of a much smaller amount of vertices, we study the maximum possible number of tree graphs which can be represented by a prescribed tree graph. We derive an algorithm for finding tight bounds for this family of problems up to chain bound discrepancy, and give upper and lower bounds in special cases.

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In this paper, we explore the singular values of adjacency matrices {An} for a particular family {Gn} of graphs, known as broom trees. The singular values of a matrix M are defined to be the square roots of the eigenvalues of the symmetrized matrix MTM. The matrices we are interested in are the symmetrized adjacency matrices AnTAn and the symmetrized exponentiated adjacency matrices BnTBn = (eAn − I)T(eAn − I) of the graphs Gn. The application of these matrices in the HITS algorithm for Internet searches suggests that we study whether the largest two eigenvalues of AnTAn (or those of BnTBn) can become close or in fact coincide. We have shown that for one family of broom-trees, the ratio of the two largest eigenvalues of BnTBn as the number n of nodes (more specifically, the length l of the graph) goes to infinity is bounded below one. This bound shows that for these graphs, the second largest eigenvalue remains bounded away from the largest eigenvalue. For a second family of broom trees it is not known whether the same is true. However, we have shown that for that family a certain later eigenvalue remains bounded away from the largest eigenvalue. Our last result is a generalization of this latter result.

Software components provide essential means to structure and organize software effectively. However, frequently, required component abstractions are not available in a programming language or system, or are not adequately combinable with each other. Invasive software composition (ISC) is a general approach to software composition that unifies component-like abstractions such as templates, aspects and macros. ISC is based on fragment composition, and composes programs and other software artifacts at the level of syntax trees. Therefore, a unifying fragment component model is related to the context-free grammar of a language to identify extension and variation points in syntax trees as well as valid component types. By doing so, fragment components can be composed by transformations at respective extension and variation points so that always valid composition results regarding the underlying context-free grammar are yielded. However, given a language’s context-free grammar, the composition result may still be incorrect. Context-sensitive constraints such as type constraints may be violated so that the program cannot be compiled and/or interpreted correctly. While a compiler can detect such errors after composition, it is difficult to relate them back to the original transformation step in the composition system, especially in the case of complex compositions with several hundreds of such steps. To tackle this problem, this thesis proposes well-formed ISC—an extension to ISC that uses reference attribute grammars (RAGs) to specify fragment component models and fragment contracts to guard compositions with context-sensitive constraints. Additionally, well-formed ISC provides composition strategies as a means to configure composition algorithms and handle interferences between composition steps. Developing ISC systems for complex languages such as programming languages is a complex undertaking. Composition-system developers need to supply or develop adequate language and parser specifications that can be processed by an ISC composition engine. Moreover, the specifications may need to be extended with rules for the intended composition abstractions. Current approaches to ISC require complete grammars to be able to compose fragments in the respective languages. Hence, the specifications need to be developed exhaustively before any component model can be supplied. To tackle this problem, this thesis introduces scalable ISC—a variant of ISC that uses island component models as a means to define component models for partially specified languages while still the whole language is supported. Additionally, a scalable workflow for agile composition-system development is proposed which supports a development of ISC systems in small increments using modular extensions. All theoretical concepts introduced in this thesis are implemented in the Skeletons and Application Templates framework SkAT. It supports “classic”, well-formed and scalable ISC by leveraging RAGs as its main specification and implementation language. Moreover, several composition systems based on SkAT are discussed, e.g., a well-formed composition system for Java and a C preprocessor-like macro language. In turn, those composition systems are used as composers in several example applications such as a library of parallel algorithmic skeletons.