Microcomputer

Issued as Reports [nos.1-2] and Final report, Project no. E-21-602 (includes subproject no. A-4431/Schlag

Another look at the "longest ascending subsequence" problem

Dijkstra has given a derivation of an efficient algorithm for a problem concerning monotonic subsequences, and extracted a proof of a related theorem from the algorithm. Here it is shown that a careful separation of concerns can lead to a beautiful conventional proof, a very different derivation of Dijkstra's algorithm, a more elegant proof from the algorithm, and the discovery of a duality property

Analyses of evolutionary algorithms

Evolutionary algorithms (EAs) are a highly successful tool commonly used in practice to solve algorithmic problems. This remarkable practical value, however, is not backed up by a deep theoretical understanding. Such an understanding would facilitate the application of EAs to further problems. Runtime analyses of EAs are one way to expand the theoretical knowledge in this field. This thesis presents runtime analyses for three prominent problems in combinatorial optimization. Additionally, it provides probability theoretical tools that will simplify future runtime analyses of EAs. The first problem considered is the Single Source Shortest Path problem. The task is to find in a weighted graph for a given source vertex shortest paths to all other vertices. Developing a new analysis method we can give tight bounds on the runtime of a previously designed and analyzed EA for this problem. The second problem is the All-Pairs Shortest Path problem. Given a weighted graph, one has to find a shortest path for every pair of vertices in the graph. For this problem we show that adding a crossover operator to a natural EA using only mutation provably decreases the runtime. This is the first time that the usefulness of a crossover operator was shown for a combinatorial problem. The third problem considered is the Sorting problem. For this problem, we design a new representation based on trees. We show that the EA nat- urally arising from this representation has a better runtime than previously analyzed EAs.Evolutionäre Algorithmen (EAs) werden in der Praxis sehr erfolgreich eingesetzt. Bisher werden die theoretischen Grundlagen von EAs jedoch nicht zufriedenstellend verstanden. Laufzeitanalysen für einfache EAs sollen dieses Verständnis erweitern. Diese Dissertation enthält Laufzeitanalysen für EAs für drei wohlbekannte kombinatorische Probleme. Zusätzlich werden wahrscheinlichkeitstheoretische Hilfsmittel zur Analyse von EAs eingeführt. Zuerst behandeln wir das Single Source Shortest Path Problem. Die Aufgabe besteht darin, in einem gewichteten Graphen einen kürzesten Weg von einem Startknoten zu jedem anderen Knoten zu finden. Durch die Entwick- lung einer neuen Analysemethode konnten wir scharfe Schranken für die Laufzeit eines bereits zuvor präsentierten und analysierten EAs angeben. Als nächstes betrachten wir das All-Pairs Shortest Path Problem. Hierbei will man für jedes Paar von Knoten in einem gewichteten Graphen einen kürzesten Weg berechnen. Für dieses Problem zeigen wir, dass das Hinzufügen eines Crossover Operators die Laufzeit gegenüber einem natürlichen EA, der nur Mutationen nutzt, verbessert. Dies ist das erste Mal, dass für ein kombinatorisches Problem bewiesen wurde, dass ein Crossover Operator die Laufzeit reduziert. Für das Sortierproblem entwickeln wir eine neue, auf Bäumen beruhende Repräsentation und zeigen, dass der natürlich daraus entstehende EA eine bessere Laufzeit hat als vorherige EAs

Combinatorial algorithms in the approximate computing paradigm

Data-intensive computing has led to the emergence of data centers with massive processor counts and main memory sizes. However, the demand for shared resources has surpassed their capacity, resulting in increased costs and limited access. Commodity hardware, although accessible, has limited computational resources. This poses a challenge when performing computationally intensive tasks with large amounts of data on systems with restricted memory. To address these issues, Approximate Computing offers a solution by allowing selective solution approximation, leading to improved resource efficiency. This dissertation focuses on the trade-off between output quality and computational resource usage in sorting and searching problems. It introduces the concept of Approximate Sorting, which aims to reduce resource usage while maintaining an accepted level of sorting quality. Quality metrics are defined to assess the ”sortedness” of approximately sorted arrays. The dissertation also proposes a general framework for incorporating approximate computing into sorting algorithms, presenting an algorithm for approximate sorting with guaranteed upper bounds. The algorithms operate under a constraint on the number of comparisons performed. The dissertation continues to explore searching algorithms, specifically binary search algorithms on approximately sorted arrays. It addresses cases where metrics are given for the input array and cases where metrics are not available. Efficient and optimal algorithms are developed for multidimensional range searches and catalog searches on approximately sorted input. The dissertation further proposes algorithms that analyze patterns in input order to optimize sorting. These algorithms identify underlying patterns and sequences, facilitating faster sorting approaches. Additionally, the dissertation discusses the growing popularity of approximate computing in the field of High-Performance Computing (HPC). It presents a novel approach to comparison-based sorting by incorporating parallel approximate computing. The dissertation also proposes algorithms for various queries on approximately sorted arrays, such as determining the rank or position of an element. The time complexity of these querying algorithms is proportional to the input metric. The dissertation concludes by emphasizing the wide range of applications for sorting and searching algorithms. In the context of packet classification in router buffers, approximate sorting offers advantages by reducing the time-consuming sorting step. By capping the number of comparisons, approximate sorting becomes a practical solution for efficiently handling the large volume of incoming packets. This dissertation contributes to the field of approximate computing by addressing resource limitations and cost issues in data-intensive computing. It provides insights into approximate sorting and searching algorithms, and their application in various domains, offering a valuable contribution to the advancement of efficient, scalable, and accessible data processing

Stochastic Optimisation Problems of Online Selection under Constraints.

PhD ThesesThis thesis deals with several closely related, but subtly di erent problems in the area of sequential stochastic optimisation. A joint property they share is the online constraint that is imposed on the decision-maker: once she observes an element, the decision whether to accept or reject it should be made immediately, without an option to recall the element in future. Observations in these problems are random variables, which take values in either R or in Rd, following known reasonably well-behaving continuous distributions. The stochastic nature of observations and the online condition shape the optimal selection policy. Furthermore, the latter indeed depends on the latest information and is updated at every step. The optimal policies may not be easily described. Even for a small number of steps, solving the optimality recursion may be computationally demanding. However, a detailed investigation yields a range of easily-constructible suboptimal policies that asymptotically perform as well as the optimal one. We aim to describe both optimal and suboptimal policies and study properties of the random processes that arise naturally in these problems. Speci cally, in this thesis we focus on the sequential selection of the longest increasing subsequence in discrete and continuous time introduced by Samuels and Steele [55], the quickest sequential selection of the increasing subsequence of a xed size recently studied by Arlotto et al. [3], and the sequential selection under a sum constraint introduced by Co man et al. [26]