Robert Preis

Linear time 1/2 -approximation algorithm for maximum weighted matching in general graphs

A new approximation algorithm for maximum weighted matching in general edge-weighted graphs is presented. It calculates a matching with an edge weight of at least 1/2 of the edge weight of a maximum weighted matching. Its time complexity is O(|E|), with |E| being the number of edges in the graph. This improves over the previously known 1/2 -approximation algorithms for maximum weighted matching which require O(|E| ċ log(|V|)) steps, where |V| is the number of vertices.

Diffusion Schemes for Load Balancing on Heterogeneous Networks

Several different diffusion schemes have previously been developed for load balancing on homogeneous processor networks. We generalize existing schemes, in order to deal with heterogeneous networks.Generalized schemes may operate efficiently on networks where each processor can have arbitrary computing power, i.e., the load will be balanced proportionally to these powers. The balancing flow that is calculated by schemes for homogeneous networks is minimal with regard to the l2 -norm and we prove this to hold true for generalized schemes, too. We demonstrate the usability of generalized schemes by a number of experiments on several heterogeneous networks.

Transport in Dynamical Astronomy and Multibody Problems

We combine the techniques of almost invariant sets (using tree structured box elimination and graph partitioning algorithms) with invariant manifold and lobe dynamics techniques. The result is a new computational technique for computing key dynamical features, including almost invariant sets, resonance regions as well as transport rates and bottlenecks between regions in dynamical systems. This methodology can be applied to a variety of multibody problems, including those in molecular modeling, chemical reaction rates and dynamical astronomy. In this paper we focus on problems in dynamical astronomy to illustrate the power of the combination of these different numerical tools and their applicability. In particular, we compute transport rates between two resonance regions for the three-body system consisting of the Sun, Jupiter and a third body (such as an asteroid). These resonance regions are appropriate for certain comets and asteroids.

Shape-optimized mesh partitioning and load balancing for parallel adaptive FEM

We present a dynamic distributed load balancing algorithm for parallel, adaptive Finite Element simulations in which we use preconditioned Conjugate Gradient solvers based on domain-decomposition. The load balancing is designed to maintain good partition aspect ratio and we show that cut size is not always the appropriate measure in load balancing. Furthermore, we attempt to answer the question why the aspect ratio of partitions plays an important role for certain solvers. We define and rate different kinds of aspect ratio and present a new center-based partitioning method of calculating the initial distribution which implicitly optimizes this measure. During the adaptive simulation, the load balancer calculates a balancing flow using different versions of the diffusion algorithm and a variant of breadth first search. Elements to be migrated are chosen according to a cost function aiming at the optimization of subdomain shapes. Experimental results for Brambles preconditioner and comparisons to state-of-the-art load balancers show the benefits of the construction.

Quality matching and local improvement for multilevel graph-partitioning

Abstract Multilevel strategies have proven to be very powerful approaches in order to partition graphs efficiently. Their efficiency is dominated by two parts; the coarsening and the local improvement strategies. Several methods have been developed to solve these problems, but their efficiency has only been proven on an experimental basis. In this paper, we present new and efficient methods for both problems, while satisfying certain quality measurements. For the coarsening part we develop a new approximation algorithm for maximum weighted matching in general edge-weighted graphs. It calculates a matching with an edge weight of at least 1 2 of the edge weight of a maximum weighted matching. Its time complexity is O(|E|), with |E| being the number of edges in the graph. Furthermore, we use the Helpful-Set strategy for the local improvement of partitions. For partitioning graphs with a regular degree of 2k into two parts, it guarantees an upper bound of ((k−1)/2)|V|+1 on the cut size of the partition, with |V| being the number of vertices. These quality methods used for the two parts of the multilevel approach lead to an efficient graph-partitioning concept.

Upper Bounds on the Bisection Width of 3- and 4-Regular Graphs

We derive new upper bounds on the bisection width of graphs which have a regular vertex degree. We show that the bisection width of large 3-regular graphs with |V| vertices is at most 1/6 |V|. For the bisection width of large 4-regular graphs we show an upper bound of 2/5 |V|.

Diffusive load balancing schemes on heterogeneous networks

Up to now, diffusive load balancing schemes have only been developed for homogeneous networks. We generalize existing diffusion schemes, in order to deal with heterogeneous networks. In these networks, every processor can have arbitrary computing power, and the load has to be balanced proportionally to these weights. The balancing flow that is calculated by the schemes for homogeneous networks is minimal with regard to the l2-norm and we prove this to hold true for the generalized schemes, too. By means of a number of experiments we demonstrate the usability of the generalized schemes on heterogeneous networks.

Satellite formation, a mobile sensor network in space

Several currently planned space missions consist of a set of satellites flying in a formation. This enables a much higher functionality of the mission compared to missions consisting of only a single satellite. On the other hand, this introduces several new problems, especially in the handling of the formation. Besides their geometric structure, the formation of satellites also has a communication network among the satellites which is the basis for the cooperative behavior, in order to accomplish the overall aim of the mission. The sensors present in each of the satellite enable sensing and communication between the satellites and form a wireless network. This gives rise to a mobile sensor network in space and hence many challenges with it. The topology of this communication in the mobile sensor network can be a bottleneck in the operation of the formation because the transmission of information and the efficient coordination of the formation relies on this topology. This is particularly the case for a large number of satellites in the network. In this paper we study the relative stability of communicating sensor topologies as a key characteristic of the formation of satellites and their behavior as a wireless sensor network in space.

Graph Algorithms for Dynamical Systems

This article is concerned with the numerical analysis of dynamical systems using methods that are based on a discretized description of the system as a graph. The graph-based description provides a unifying framework to approach a wide and diverse variety of dynamical systems, from time-discrete maps via ordinary differential equations to stochastic differential equations describing e. g. diffusion in a potential landscape. Within this variety, this article focusses on those dynamical systems that can possess a ‘multiscale structure’ in the sense that they exhibit interesting dynamical behavior on more than one timescale. We will explain what we mean with this phrase by means of some examples. Consider in Fig. 1 one trajectory of Chua’s circuit, that is described by the well-known threedimensional ordinary differential equation

Upper bounds on the bisection width of 3- and 4-regular graphs

Abstract We derive new upper bounds on the bisection width of graphs which have a regular vertex degree. We show that the bisection width of sufficiently large 3-regular graphs with | V | vertices is at most ( 1 6 + e ) | V | , e > 0 . For the bisection width of sufficiently large 4-regular graphs we show an upper bound of ( 2 5 + e ) | V | , e > 0 .