Symmetric Interconnection Networks from Cubic Crystal Lattices

Torus networks of moderate degree have been widely used in the supercomputer industry. Tori are superb when used for executing applications that require near-neighbor communications. Nevertheless, they are not so good when dealing with global communications. Hence, typical 3D implementations have evolved to 5D networks, among other reasons, to reduce network distances. Most of these big systems are mixed-radix tori which are not the best option for minimizing distances and efficiently using network resources. This paper is focused on improving the topological properties of these networks. By using integral matrices to deal with Cayley graphs over Abelian groups, we have been able to propose and analyze a family of high-dimensional grid-based interconnection networks. As they are built over $n$-dimensional grids that induce a regular tiling of the space, these topologies have been denoted \textsl{lattice graphs}. We will focus on cubic crystal lattices for modeling symmetric 3D networks. Other higher dimensional networks can be composed over these graphs, as illustrated in this research. Easy network partitioning can also take advantage of this network composition operation. Minimal routing algorithms are also provided for these new topologies. Finally, some practical issues such as implementability and preliminary performance evaluations have been addressed

Gray codes and their applications

An n-bit Gray code is an ordered set of all 2n binary strings of length n. The special property of this listing is that Hamming distance between consecutive vectors is exactly 1. If the last and first codeword also have a Hamming distance 1 then the code is said to be cyclic. This dissertation addresses problems dealing with the design and applications of new and existing types of both binary and non-binary Gray codes. It is shown how properties of certain Gray codes can be used to solve problems arising in different domains. New types of Gray codes to solve specific types of problems are also designed. We construct Gray codes over higher integral radices and show their applications. Applications of new classes of Gray codes defined over residue classes of Gaussian integers are also shown. We also propose new classes of binary Gray codes and prove some important properties of these codes

Extremal Spectral Dynamics and a Fractal Theory for Simplicial Complexes

The aim of this work is the exploration of spectral asymptotics of certain geometries associated to simplicial complexes. We will state how combinatorial and differential Laplacians can be associated to a simplicial complex and describe certain asymptotics linked to their spectra. First we take under consideration the change of spectrum for the combinatorial Laplacian under a certain class of subdivision procedures and show a universal limit theorem regarding the sequence arising from this construction. Universality in this case means that the limit spectrum carries no spectral information related to the input complex. It is thus only dependent on the dimension of the complex and the subdivision procedure used. We will carry out the explicit calculation of such a limit for one particular example of a subdivision related to barycentric subdivision. Next we point out obstructions to the application of the same procedure to the full barycentric subdivision. It will turn out that the procedure is not favorable if the given subdivision procedure is acting non-trivially on lower dimensional faces. Lastly we give an example of a subdivision procedure of high symmetry, i.e. edgewise subdivision, for which we can determine the spectrum by a group action argument even though it acts non-trivially on lower dimensional faces. Furthermore dual relations to fractal theory are examined and the particular class of fractals arising from subdivision of a complex in the sense of a graph-directed limit construction is formalized. In the end open question regarding the nature of the limit are formulated and initiating thoughts are presented. Secondly we associate to a simplicial complex a geometry (not necessarily embeddable in euclidean space) and show that there exists a natural differential Laplacian on this geometry. These complexes can be used to model thin structures around their geometry. As this modelling procedure is a higher-dimensional generalization of quantum graphs we will call a complex equipped with this differential structure a quantum complex. Thin structures over such a complex allow for modelling of systems with a larger number of dimensions not constraint by a small diameter. We show generalizations of estimates used in the proof of the spectral asymptotic of these thin structures for the graph case indicating that a general spectral asymptotic might be possible. We formulate open questions on spectral asymptotics and the relation of the combinatorial and differential Laplacian associated to the complex

Proceedings of the 3rd International Workshop on Optimal Networks Topologies IWONT 2010

Peer Reviewe

Applications of transfer operator methods to the dynamics of low-dimensional piecewise smooth maps

This thesis primarily concentrates on stochastic and spectral properties of the transfer operator generated by piecewise expanding maps (PWEs) and piecewise isometries (PWIs). We also consider the applications of the transfer operator in thermodynamic formalism. The original motivation stems from studies of one-dimensional PWEs. In particular, any one dimensional mixing PWE admits a unique absolutely continuous invariant probability measure (ACIP) and this ACIP has a bounded variation density. The methodology used to prove the existence of this ACIP is based on a so-called functional analytic approach and a key step in this approach is to show that the corresponding transfer operator has a spectral gap. Moreover, when a PWE has Markov property this ACIP can also be viewed as a Gibbs measure in thermodynamic formalism. In this thesis, we extend the studies on one-dimensional PWEs in several aspects. First, we use the functional analytic approach to study piecewise area preserving maps (PAPs) in particular to search for the ACIPs with multidimensional bounded variation densities. We also explore the relationship between the uniqueness of ACIPs with bounded variation densities and topological transitivity/ minimality for PWIs. Second, we consider the mixing and corresponding mixing rate properties of a collection of piecewise linear Markov maps generated by composing x to mx mod 1 with permutations in SN. We show that typical permutations preserve the mixing property under the composition. Moreover, by applying the Fredholm determinant approach, we calculate the mixing rate via spectral gaps and obtain the max/min spectral gaps when m,N are fixed. The spectral gaps can be made arbitrarily small when the permutations are fully refined. Finally, we consider the computations of fractal dimensions for generalized Moran constructions, where different iteration function systems are applied on different levels. By using the techniques in thermodynamic formalism, we approximate the fractal dimensions via the zeros of the Bowen's equation on the pressure functions truncated at each level