Subcube embeddability and fault tolerance of augmented hypercubes

Hypercube networks have received much attention from both parallel processing and communications areas over the years since they offer a rich interconnection structure with high bandwidth, logarithmic diameter, and high degree of fault tolerance. They are easily partitionable and exhibit a high degree of fault tolerance. Fault-tolerance in hypercube and hypercube-based networks received the attention of several researchers in recent years; The primary idea of this study is to address and analyze the reliability issues in hypercube networks. It is well known that the hypercube can be augmented with one dimension to replace any of the existing dimensions should any dimension fail. In this research, it is shown that it is possible to add i dimensions to the standard hypercube, Qn to tolerate (i - 1) dimension failures, where 0 \u3c i ≤ n. An augmented hypercube, Qn +(n) with n additional dimensions is introduced and compared with two other hypercube networks with the same amount of redundancy. Reliability analysis for the three hypercube networks is done using the combinatorial and Markov modeling. The MTTF values are calculated and compared for all three networks. Comparison between similar size hypercube networks show that the augmented hypercube is more robust than the standard hypercube; As a related problem, we also look at the subcube embeddability. Subcube embeddability of the hypercube can be enhanced by introducing an additional dimension. A set of new dimensions, characterized by the Hamming distance between the pairs of nodes it connects, is introduced using a measure defined as the magnitude of a dimension. An enumeration of subcubes of various sizes is presented for a dimension parameterized by its magnitude. It is shown that the maximum number of subcubes for a Qn can only be attained when the magnitude of dimension is n - 1 or n. It is further shown that the latter two dimensions can optimally increase the number of subcubes among all possible choices

Neural Relax

We present an algorithm for data preprocessing of an associative memory inspired to an electrostatic problem that turns out to have intimate relations with information maximization

A general analytical model of adaptive wormhole routing in k-ary n-cubes

Several analytical models of fully adaptive routing have recently been proposed for k-ary n-cubes and hypercube networks under the uniform traffic pattern. Although,hypercube is a special case of k-ary n-cubes topology, the modeling approach for hypercube is more accurate than karyn-cubes due to its simpler structure. This paper proposes a general analytical model to predict message latency in wormhole-routed k-ary n-cubes with fully adaptive routing that uses a similar modeling approach to hypercube. The analysis focuses Duato's fully adaptive routing algorithm [12], which is widely accepted as the most general algorithm for achieving adaptivity in wormhole-routed networks while allowing for an efficient router implementation. The proposed model is general enough that it can be used for hypercube and other fully adaptive routing algorithms

Real-valued feature selection for process approximation and prediction

The selection of features for classification, clustering and approximation is an important task in pattern recognition, data mining and soft computing. For real-valued features, this contribution shows how feature selection for a high number of features can be implemented using mutual in-formation. Especially, the common problem for mutual information computation of computing joint probabilities for many dimensions using only a few samples is treated by using the Rènyi mutual information of order two as computational base. For this, the Grassberger-Takens corre-lation integral is used which was developed for estimating probability densities in chaos theory. Additionally, an adaptive procedure for computing the hypercube size is introduced and for real world applications, the treatment of missing values is included. The computation procedure is accelerated by exploiting the ranking of the set of real feature values especially for the example of time series. As example, a small blackbox-glassbox example shows how the relevant features and their time lags are determined in the time series even if the input feature time series determine nonlinearly the output. A more realistic example from chemical industry shows that this enables a better ap-proximation of the input-output mapping than the best neural network approach developed for an international contest. By the computationally efficient implementation, mutual information becomes an attractive tool for feature selection even for a high number of real-valued features