Resource placement, data rearrangement, and Hamiltonian cycles in torus networks

Many parallel machines, both commercial and experimental, have been/are being designed with toroidal interconnection networks. For a given number of nodes, the torus has a relatively larger diameter, but better cost/performance tradeoffs, such as higher channel bandwidth, and lower node degree, when compared to the hypercube. Thus, the torus is becoming a popular topology for the interconnection network of a high performance parallel computers. In a multicomputer, the resources, such as I/O devices or software packages, are distributed over the networks. The first part of the thesis investigates efficient methods of distributing resources in a torus network. Three classes of placement methods are studied. They are (1) distant-t placement problem: in this case, any non-resource node is at a distance of at most t from some resource nodes, (2) j-adjacency problem: here, a non-resource node is adjacent to at least j resource nodes, and (3) generalized placement problem: a non-resource node must be a distance of at most t from at least j resource nodes. This resource placement technique can be applied to allocating spare processors to provide fault-tolerance in the case of the processor failures. Some efficient spare processor placement methods and reconfiguration schemes in the case of processor failures are also described. In a torus based parallel system, some algorithms give best performance if the data are distributed to processors numbered in Cartesian order; in some other cases, it is better to distribute the data to processors numbered in Gray code order. Since the placement patterns may be changed dynamically, it is essential to find efficient methods of rearranging the data from Gray code order to Cartesian order and vice versa. In the second part of the thesis, some efficient methods for data transfer from Cartesian order to radix order and vice versa are developed. The last part of the thesis gives results on generating edge disjoint Hamiltonian cycles in k-ary n-cubes, hypercubes, and 2D tori. These edge disjoint cycles are quite useful for many communication algorithms

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

Proceedings of the First International Workshop on Mathematical Foundations of Computational Anatomy (MFCA'06) - Geometrical and Statistical Methods for Modelling Biological Shape Variability

International audienceNon-linear registration and shape analysis are well developed research topic in the medical image analysis community. There is nowadays a growing number of methods that can faithfully deal with the underlying biomechanical behaviour of intra-subject shape deformations. However, it is more difficult to relate the anatomical shape of different subjects. The goal of computational anatomy is to analyse and to statistically model this specific type of geometrical information. In the absence of any justified physical model, a natural attitude is to explore very general mathematical methods, for instance diffeomorphisms. However, working with such infinite dimensional space raises some deep computational and mathematical problems. In particular, one of the key problem is to do statistics. Likewise, modelling the variability of surfaces leads to rely on shape spaces that are much more complex than for curves. To cope with these, different methodological and computational frameworks have been proposed. The goal of the workshop was to foster interactions between researchers investigating the combination of geometry and statistics for modelling biological shape variability from image and surfaces. A special emphasis was put on theoretical developments, applications and results being welcomed as illustrations. Contributions were solicited in the following areas: * Riemannian and group theoretical methods on non-linear transformation spaces * Advanced statistics on deformations and shapes * Metrics for computational anatomy * Geometry and statistics of surfaces 26 submissions of very high quality were recieved and were reviewed by two members of the programm committee. 12 papers were finally selected for oral presentations and 8 for poster presentations. 16 of these papers are published in these proceedings, and 4 papers are published in the proceedings of MICCAI'06 (for copyright reasons, only extended abstracts are provided here)