Crosslinking in parallel

A crosslink is a double link established between the two entries of an edge in an adjacency list representation of a graph. Crosslinks play important roles in several parallel algorithms as they provide constant time access between the two entries of an edge; the existence of crosslinks is usually assumed. We consider the problem of establishing crosslinks in a crosslink-less adjacency list for graphs that belong to a class of graphs called the linearly contractible graphs, and show that cross-links can be established optimally in O(log n log*n) time using a CREW PRAM and optimally in O(log n) time using a CRCW PRAM for such graphs

The Flip Diameter of Rectangulations and Convex Subdivisions

We study the configuration space of rectangulations and convex subdivisions of $n$ points in the plane. It is shown that a sequence of $O(n\log n)$ elementary flip and rotate operations can transform any rectangulation to any other rectangulation on the same set of $n$ points. This bound is the best possible for some point sets, while $\Theta(n)$ operations are sufficient and necessary for others. Some of our bounds generalize to convex subdivisions of $n$ points in the plane.Comment: 17 pages, 12 figures, an extended abstract has been presented at LATIN 201

Non-Monochromatic and Conflict-Free Coloring on Tree Spaces and Planar Network Spaces

It is well known that any set of n intervals in $\mathbb{R}^1$ admits a non-monochromatic coloring with two colors and a conflict-free coloring with three colors. We investigate generalizations of this result to colorings of objects in more complex 1-dimensional spaces, namely so-called tree spaces and planar network spaces

Exact Algorithms for NP-hard Problems on Networks: Design, Analysis, and Implementation

Die Arbeit befasst sich mit dem Entwurf von exakten Algorithmen für verschiedene NP-vollständige Optimierungsprobleme auf Graphen, wie etwa Vertex Cover, Independent Set, oder Dominating Set. Viele praxisbezogene Aufgaben, beispielsweise sogenannte 'facility location'-Probleme aus dem Bereich der Entscheidungsanalyse (decision analysis), sind durch entsprechende Netzwerk-Modellierung auf derartige Fragestellungen zurückzuführen. Im Vordergrund der Arbeit stehen Lösungsverfahren mit beweisbaren Laufzeitschranken. Wegen der solchen Problemen inhärenten, hohen kombinatorischen Komplexität müssen wir exponentielles Laufzeitverhalten unserer Algorithmen in Kauf nehmen, wollen dieses jedoch kleinstmöglich halten. Wir verfolgen dabei den jüngst vorgeschlagenen Ansatz sogenannter 'parametrisierter Algorithmen'. Vereinfacht gesagt handelt es sich hierbei um eine zweidimensionale Herangehensweise, bei welcher die Laufzeit nicht ausschlie3lich in der Grö3e der Eingabeinstanz, sondern überdies auch in der Grö3e eines sogenannten 'Problemparameters' gemessen wird. Dabei untersuchen wir sowohl von theoretischer, als auch von praktischer Seite unterschiedliche Methoden des Algorithmen-Designs: Datenreduktion, beschränkte Suchbäume, Separation von Graphen und das Konzept von Baumzerlegungen. Schlie3lich stellen wir ein Software-Paket vor, welches im Rahmen dieses Projektes entwickelt wurde und eine Vielzahl der entwickelten Algorithmen implementiert. Wir berichten über eine Reihe von empirischen Studien zur Auswertung der Praxistauglichkeit dieser Algorithmen.This thesis deals with the design of exact algorithms for various NP-complete optimization problems on graphs like Vertex Cover, Independent Set, or Dominating Set. We encounter such problems in a broad variety of application ranges, e.g., when modelling so-called facility location tasks in the area of decision analysis and network design. The main focus of this work is on solving algorithms with provable bounds on the running time. Due to the seemingly unavoidable inherent high combinatorial complexity of the problems under consideration, we are forced to deal with exponential running times; our goal, however, is to keep this exponential part as low as possible. To this end, we follow a recent approach of so-called 'fixed-parameter algorithms,' where the running time of an algorithm solving this problem shall be measured not only in the size of the input instance, but also in the size of a so-called 'problem parameter.' In this sense, whereas classical complexity theory offers a one-dimensional approach, parameterized complexity theory is a two-dimensional study of combinatorial problems. We investigate from both, a theoretical, as well as a practical point of view, various methods in the design of fixed-parameter algorithms: data reduction, bounded search trees, graph separation, and the concept of tree-decompositions. In addition, a software-package is presented, which was developed in our project and which implements most of our algorithms. Finally, we report on a first series of empirical studies underpinning the practical strength and usefulness of our algorithms