Binary Hypercube Research Articles

Optimal Local Identifying and Local Locating-dominating Codes

We introduce two new classes of covering codes in graphs for every positive integer r. These new codes are called local r-identifying and local r-locating-dominating codes and they are derived from r-identifying and r-locating-dominating codes, respectively. We study the sizes of optimal local 1-identifying codes in binary hypercubes. We obtain lower and upper bounds that are asymptotically tight. Together the bounds show that the cost of changing covering codes into local 1-identifying codes is negligible. For some small n optimal constructions are obtained. Moreover, the upper bound is obtained by a linear code construction. Also, we study the densities of optimal local 1-identifying codes and local 1-locating-dominating codes in the infinite square grid, the hexagonal grid, the triangular grid and the king grid. We prove that seven out of eight of our constructions have optimal densities.

Reliability Evaluation of Generalized Exchanged Hypercubes Based on Imprecise Diagnosis Strategies

Fault diagnostic analysis is extremely important for interconnection networks. The [Formula: see text]-diagnosis imprecise strategy plays an essential role in the reliability of networks. The [Formula: see text]-diagnosis strategy can detect up to [Formula: see text] faulty vertices which might include at most [Formula: see text] misdiagnosed vertices. The exchanged hypercube is obtained by systematically removing links from a binary hypercube, which has smaller maximum degree and Wiener index than the hypercube. We use [Formula: see text] to denote the generalized exchanged hypercube, and show in this paper that [Formula: see text] is [Formula: see text]-diagnosable with [Formula: see text] and [Formula: see text] under the PMC model and MM[Formula: see text] model. We also propose a [Formula: see text]-diagnosis algorithm on [Formula: see text]. As a side benefit, the [Formula: see text]-diagnosability of the dual-cube-like network [Formula: see text] can be directly obtained from our results.

Minimum layout of hypercubes and folded hypercubes into the prism over caterpillars

The problem of embedding an n-node graph G into an n-node graph H is an important aspect in parallel and distributed processing. Graph embedding results have been successfully used to establish equivalence of interconnections in parallel and distributed machines. The binary hypercube is one of the most widely used multiprocessor systems due to its simple, deadlock-free routing and broadcasting properties. The folded hypercube is an important variant of hypercube with the same number of nodes. Trees are the fundamental graph theoretical models in many fields including artificial intelligence and various network designs. In this paper, we consider the problem of embedding the hypercube and folded hypercube into the prism over a caterpillar in such a way as to minimise its layout.

Node set optimization problem for complete Josephus cubes

The problem of finding an optimal node set in an interconnection network plays an important role in minimizing the layout of embedding the network into linear chassis. In this paper we find the nested optimal node sets for a complete Josephus cube, a recently proposed fault-tolerant node cluster architecture variant of the binary hypercube which has the same number of nodes as the hypercube but exhibits enhanced embedding, fault tolerance and communications performance than the hypercube and many of its variants. As a byproduct we obtain the minimum layout of embedding the complete Josephus cube into a path, 1-rooted complete binary tree, sibling tree and caterpillar.

Exact recovery in the Ising blockmodel

We consider the problem associated to recovering the block structure of an Ising model given independent observations on the binary hypercube. This new model, called the Ising blockmodel, is a perturbation of the mean field approximation of the Ising model known as the Curie–Weiss model: the sites are partitioned into two blocks of equal size and the interaction between those of the same block is stronger than across blocks, to account for more order within each block. We study probabilistic, statistical and computational aspects of this model in the high-dimensional case when the number of sites may be much larger than the sample size.

Improved log-Sobolev inequalities, hypercontractivity and uncertainty principle on the hypercube

Log-Sobolev inequalities (LSIs) upper-bound entropy via a multiple of the Dirichlet form (i.e. norm of a gradient). In this paper we prove a family of entropy-energy inequalities for the binary hypercube which provide a non-linear comparison between the entropy and the Dirichlet form and improve on the usual LSIs for functions with small support. These non-linear LSIs, in turn, imply a new version of the hypercontractivity for such functions. As another consequence, we derive a sharp form of the uncertainty principle for the hypercube: a function whose energy is concentrated on a set of small size, and whose Fourier energy is concentrated on a small Hamming ball must be zero. The tradeoff between the sizes that we derive is asymptotically optimal. This new uncertainty principle implies a new estimate on the size of Fourier coefficients of sparse Boolean functions. We observe that an analogous (asymptotically optimal) uncertainty principle in the Euclidean space follows from the sharp form of Young's inequality due to Beckner. This hints that non-linear LSIs augment Young's inequality (which itself is sharp for finite groups).

List-Decodable Zero-Rate Codes

We consider list decoding in the zero-rate regime for two cases—the binary alphabet and the spherical codes in Euclidean space. Specifically, we study the maximal $\tau \in [{0,1}]$ for which there exists an arrangement of $M$ balls of relative Hamming radius $\tau $ in the binary hypercube (of arbitrary dimension) with the property that no point of the latter is covered by $L$ or more of them. As $M\to \infty $ the maximal $\tau $ decreases to a well-known critical value $\tau _{L}$ . In this paper, we prove several results on the rate of this convergence. For the binary case, we show that the rate is $\Theta (M^{-1})$ when $L$ is even, thus extending the classical results of Plotkin and Levenshtein for $L=2$ . For $L=3$ , the rate is shown to be $\Theta (M^{-({2}/{3})})$ . For the similar question about spherical codes, we prove the rate is $\Omega (M^{-1})$ and $O(M^{-({2L}/{L^{2}-L+2})})$ .

Hypercontractivity of Spherical Averages in Hamming Space

Consider the linear space of functions on the binary hypercube and the linear operator $S_\delta$ acting by averaging a function over a Hamming sphere of radius $\delta n$ around every point. It is...

A linear time algorithm for embedding locally twisted cube into grid network to optimize the layout

Given a guest graph G and a host graph H, the layout of an embedding from G into H is to find the sum of lengths of all the shortest paths in H corresponding to the edges in G. In this paper, we present an algorithm to embed a variant of the binary hypercube known as the locally twisted cube into a 2-dimensional grid architecture and obtain the exact layout of the embedding in linear time.

$q$-counting hypercubes in Lucas cubes

Lucas and Fibonacci cubes are special subgraphs of the binary hypercubes that have been proposed as models of interconnection networks. Since these families are closely related to hypercubes, it is natural to consider the nature of the hypercubes they contain. Here we study a generalization of the enumerator polynomial of the hypercubes in Lucas cubes, which $q$-counts them by their distance to the all 0 vertex. Thus, our bivariate polynomials refine the count of the number of hypercubes of a given dimension in Lucas cubes and for $q=1$ they specialize to the cube polynomials of Klavžar and Mollard. We obtain many properties of these polynomials as well as the $q$-cube polynomials of Fibonacci cubes themselves. These new properties include divisibility, positivity, and functional identities for both families.

Layout of embedding locally twisted cube into the extended theta mesh topology

A graph embedding is an important aspect in establishing equivalence of interconnections in parallel and distributed machines. Finding the minimum layout serves as a cost criterion for evaluating a good embedding. The binary hypercube is one of the most widely used topology for interconnection networks due to its simple, deadlock-free routing and broadcasting properties. The locally twisted cube is an important variant of the hypercube with the same number of vertices and connections but exhibits enhanced properties than its counterpart. In this paper we consider the problem of embedding an n-dimensional locally twisted cube into the extended rooted theta mesh in such a way as to minimize its layout.

Unoriented first-passage percolation on the n-cube

The $n$-dimensional binary hypercube is the graph whose vertices are the binary $n$-tuples $\{0,1\}^{n}$ and where two vertices are connected by an edge if they differ at exactly one coordinate. We prove that if the edges are assigned independent mean 1 exponential costs, the minimum length $T_{n}$ of a path from $(0,0,\dots,0)$ to $(1,1,\dots,1)$ converges in probability to $\ln(1+\sqrt{2})\approx0.881$. It has previously been shown by Fill and Pemantle [Ann. Appl. Probab. 3 (1993) 593–629] that this so-called first-passage time asymptotically almost surely satisfies $\ln(1+\sqrt{2})-o(1)\leq T_{n}\leq1+o(1)$, and has been conjectured to converge in probability by Bollobás and Kohayakawa [In Combinatorics, Geometry and Probability (Cambridge, 1993) (1997) 129–137 Cambridge]. A key idea of our proof is to consider a lower bound on Richardson’s model, closely related to the branching process used in the article by Fill and Pemantle to obtain the bound $T_{n}\geq\ln(1+\sqrt{2})-o(1)$. We derive an explicit lower bound on the probability that a vertex is infected at a given time. This result is formulated for a general graph and may be applicable in a more general setting.

Wirelength of Enhanced Hypercubes into r-Rooted Complete Binary Trees

One of the central issues in designing and evaluating an interconnection network is to find out its ability to execute parallel algorithms developed for one network into another with minimum time delay. Solving the wirelength problem helps in minimizing the time delay in execution. The binary hypercube is one of the most popular interconnection networks since it has a simple structure and is easy to implement. The enhanced hypercube is an important variant of hypercube with a smaller diameter, improvised mean node distance and cost efficiency when compared to a binary hypercube. In this paper we consider the problem of embedding enhanced hypercubes into r-rooted complete binary trees for minimizing the wirelength.

Learning iterative quantization binary codes for face recognition

Binary feature descriptors have been widely used in computer vision field due to their excellent discriminative power and strong robustness, and local binary patterns (LBP) and its variations have proven that they are effective face descriptors. However, the forms of such binary feature descriptors are predefined in the hand-crafted way, which requires strong domain knowledge to design them. In this paper, we propose a simple and efficient iterative quantization binary codes (IQBC) feature learning method to learn a discriminative binary face descriptor in the data-driven way. Firstly, similar to traditional LBP method, we extract patch-wise pixel difference vectors (PDVs) by computing and concatenating the difference between center patch and its neighboring patches. Then, inspired by multi-class spectral clustering and the orthogonal Procrustes problem, which both are widely used in image retrieval field, we learn an optimized rotation to minimize the quantization error of mapping data to the vertices of a zero-centered binary hypercube by using iterative quantization scheme. In other words, we learn a feature mapping to project these pixel difference vectors into low-dimensional binary vectors. And our IQBC can be used with unsupervised data embedding method such as principle component analysis (PCA) and supervised data embedding method such as canonical correlation analysis (CCA), namely IQBC-PCA and IQBC-CCA. Lastly, we cluster and pool these projected binary codes into a histogram-based feature that describes the co-occurrence of binary codes. And we consider the histogram-based feature as our final feature representation for each face image. We investigate the performance of our IQBC-PCA and IQBC-CCA on FERET, CAS-PEAL-R1, LFW and PaSC databases. Extensive experimental results demonstrate that our IQBC descriptor outperforms other state-of-the-art face descriptors.

The metric dimension for resolving several objects

A set of vertices S is a resolving set in a graph if each vertex has a unique array of distances to the vertices of S. The natural problem of finding the smallest cardinality of a resolving set in a graph has been widely studied over the years. In this paper, we wish to resolve a set of vertices (up to ℓ vertices) instead of just one vertex with the aid of the array of distances. The smallest cardinality of a set S resolving at most ℓ vertices is called ℓ-set-metric dimension. We study the problem of the ℓ-set-metric dimension in two infinite classes of graphs, namely, the two dimensional grid graphs and the n-dimensional binary hypercubes.

An Analysis of Hypermesh NoCs in FPGAs

Accurate analytic models for the area, delay and power of the Hypermesh NoC topology, realized with the Altera family of FPGAs, are presented. Hypermeshes are based on the concept of hypergraphs, which consist of a set of nodes and a set of hyperedges, where the hyperedges represent low-latency switches which interconnect multiple nodes with deterministic latencies. Three different switch designs for the hyperedges are proposed and evaluated. Two parallel algorithms are considered; (a) the Bitonic sorting algorithm, and (b) the FFT parallel algorithm. The analytic models are shown to be very accurate, typically within 6 percent. The 2D Hypermesh is compared to the 2D layouts of the binary hypercubes (BHC) and generalized hypercubes (GHC) in terms of area, energy per algorithm, and the Energy-Area product . The Energy-Area product is proposed as an useful design metric to evaluate NoCs, which combines both the cost and the performance metrics of an NoC into one. Our analysis indicates that the 2D Hypermeshes generally have considerably lower area, energy, and Energy-Area product compared to the 2D layouts of the Hypercubes.

Fault-tolerant Routing in Dual-cubes Based on Routing Probabilities

Abstract Recently, there has been a growing demand for large-scale computing in various fields. A binary hypercube has been used as an interconnection network topology in parallel computers for large-scale computing. However, the hypercube has a limitation that the degree, that is, the number of links for each node augments rapidly according to the increase in the number of total nodes. This limitation has been complemented in a dual-cube. In this paper, we propose a fault-tolerant routing algorithm in a dual-cube based on routing probabilities. The routing probability is developed as limited global information for routing ability of a node in a hypercube for an arbitrary node located at a specific distance. Unlike other previous researches that use limited global information in each cluster on fault tolerant routing in a dual-cube, our routing algorithm applies limited global information for an overall dual-cube. Each node selects a neighbor node to forward the message to the destination node by considering routing probabilities of its neighbor nodes. The results of a computer experiment show better performance of our algorithm.

On the existence of accessible paths in various models of fitness landscapes

We present rigorous mathematical analyses of a number of well-known mathematical models for genetic mutations. In these models, the genome is represented by a vertex of the $n$-dimensional binary hypercube, for some $n$, a mutation involves the flipping of a single bit, and each vertex is assigned a real number, called its fitness, according to some rules. Our main concern is with the issue of existence of (selectively) accessible paths; that is, monotonic paths in the hypercube along which fitness is always increasing. Our main results resolve open questions about three such models, which in the biophysics literature are known as house of cards (HoC), constrained house of cards (CHoC) and rough Mount Fuji (RMF). We prove that the probability of there being at least one accessible path from the all-zeroes node $\mathbf{v}^{0}$ to the all-ones node $\mathbf{v}^{1}$ tends respectively to $0$, $1$ and $1$, as $n$ tends to infinity. A crucial idea is the introduction of a generalization of the CHoC model, in which the fitness of $\mathbf{v} ^{0}$ is set to some $\alpha=\alpha_{n}\in[0,1]$. We prove that there is a very sharp threshold at $\alpha_{n}=\frac{\ln n}{n}$ for the existence of accessible paths from $\mathbf{v}^{0}$ to $\mathbf{v}^{1}$. As a corollary we prove significant concentration, for $\alpha$ below the threshold, of the number of accessible paths about the expected value (the precise statement is technical; see Corollary 1.4). In the case of RMF, we prove that the probability of accessible paths from $\mathbf{v}^{0}$ to $\mathbf{v}^{1}$ existing tends to $1$ provided the drift parameter $\theta=\theta_{n}$ satisfies $n\theta_{n}\rightarrow\infty$, and for any fitness distribution which is continuous on its support and whose support is connected.

Iterative Quantization: A Procrustean Approach to Learning Binary Codes for Large-Scale Image Retrieval

This paper addresses the problem of learning similarity-preserving binary codes for efficient similarity search in large-scale image collections. We formulate this problem in terms of finding a rotation of zero-centered data so as to minimize the quantization error of mapping this data to the vertices of a zero-centered binary hypercube, and propose a simple and efficient alternating minimization algorithm to accomplish this task. This algorithm, dubbed iterative quantization (ITQ), has connections to multiclass spectral clustering and to the orthogonal Procrustes problem, and it can be used both with unsupervised data embeddings such as PCA and supervised embeddings such as canonical correlation analysis (CCA). The resulting binary codes significantly outperform several other state-of-the-art methods. We also show that further performance improvements can result from transforming the data with a nonlinear kernel mapping prior to PCA or CCA. Finally, we demonstrate an application of ITQ to learning binary attributes or "classemes" on the ImageNet data set.

Evolutionary Accessibility of Modular Fitness Landscapes

A fitness landscape is a mapping from the space of genetic sequences, which is modeled here as a binary hypercube of dimension $L$, to the real numbers. We consider random models of fitness landscapes, where fitness values are assigned according to some probabilistic rule, and study the statistical properties of pathways to the global fitness maximum along which fitness increases monotonically. Such paths are important for evolution because they are the only ones that are accessible to an adapting population when mutations occur at a low rate. The focus of this work is on the block model introduced by A.S. Perelson and C.A. Macken [Proc. Natl. Acad. Sci. USA 92:9657 (1995)] where the genome is decomposed into disjoint sets of loci (`modules') that contribute independently to fitness, and fitness values within blocks are assigned at random. We show that the number of accessible paths can be written as a product of the path numbers within the blocks, which provides a detailed analytic description of the path statistics. The block model can be viewed as a special case of Kauffman's NK-model, and we compare the analytic results to simulations of the NK-model with different genetic architectures. We find that the mean number of accessible paths in the different versions of the model are quite similar, but the distribution of the path number is qualitatively different in the block model due to its multiplicative structure. A similar statement applies to the number of local fitness maxima in the NK-models, which has been studied extensively in previous works. The overall evolutionary accessibility of the landscape, as quantified by the probability to find at least one accessible path to the global maximum, is dramatically lowered by the modular structure.