Optimal Hypercube Research Articles

Labeling-Based Recipient Identification with Low-Order Modulation

Labeling-Based Recipient Identification (LABRID) brings the possibility of representing the destination station address in a Bit-Interleaved Coded Modulation with Iterative Decoding (BICM-ID) system by a signal labeling rule. Low-order modulations, such as BPSK or QPSK, pose a general problem for BICM-ID due to a limited convergence of iterative decoding. In the context of LABRID, they have one more drawback—a small number of different labeling rules in general; the number of the optimal ones, which exhibit the maximal asymptotic coding gain, is reduced even further. Meanwhile, LABRID needs a sizable collection of different optimal labeling rules to serve many users in large wireless networks. In this paper, the author suggests the use of hypercube BPSK or QPSK labeling to overcome all these challenges. By means of the Reactive Tabu Search (RTS) algorithm, more than 1500 equivalent optimal hypercube labeling rules are found. Analytical error bounds of the system are developed and supported by simulation experiments. Then, the focus is moved to the criterion to determine the frame destination at the LABRID receiver; a simple threshold-based method is proposed to keep the incorrect decision probability below 10−5. Finally, it is shown that LABRID outperforms a reference BICM-ID system in terms of computational complexity.

A Computational Fluid Dynamics-based Correlation to Predict Droplet Sauter Mean Diameter for Σ− Yliq Atomization Model in Pressure Swirl Atomizer

Liquid atomization is crucial to ensure efficient combustion as it is an inherent part of the injector system. The combustion of fuels relies on effective atomization to increase the surface area of the fuel and consequently achieve high rates of mixing and evaporation. Pressure swirl atomizers are inexpensive and reliable type of atomizer for fuel injection owing to its superior atomization characteristics and relatively simple geometry. The Sauter mean diameter (SMD) of atomizer contributes significantly to the combustion chamber performance. This paper presents a two-step strategy to predict droplet SMD for atomisation model in pressure swirl atomizer through the combination of experimentally validated Computation Fluid Dynamics (CFD) and Optimal Latin Hypercubes (OLHC) Design of Experiments (DoE) techniques. A three-dimensional Eulerian two-phase CFD model is developed to account for liquid and gas phases as a single continuum with high-density variation at large Reynolds and Weber numbers and validated against experimental measurements, before being employed to carry out a parametric study involving operating conditions and fluid properties of the pressure swirl atomizer. The atomizer is then represented in terms of four design variables, namely liquid viscosity, liquid velocity, surface tension and atomizer exit diameter. An 87-point OLHC DoE is constructed within the design variables space using a permutation genetic algorithm resulting in an accurate SMD prediction. Results show the newly developed SMD prediction is found to be superior compared with existing correlations and indicate significant improvement in the droplets SMD.

Parametric exploration on the airfoil design space by numerical design of experiment methodology and multiple regression model

Robust airfoil design is crucial to efficient, stable, and safe operation for modern wind turbines. However, even for deterministic wind turbine airfoil design, the problem is complex regarding to aerodynamic, acoustic, and structural requirements of wind turbine blades. Therefore, this study aims to assess the design variable impact, identify significant variables, and obtain the correlation with the airfoil responses, to reduce the cost of the airfoil robust optimization. In this paper, the optimal hypercube design method was applied to an airfoil designed by the National Advisory Committee for Aeronautics, NACA 63-421, which is commonly employed in the outboard modern wind turbine blade, to perform the numerical design of experiments. Then, a parametric exploration on the characteristics of airfoil design space by the multiple regression model and statistical analysis method were conducted. It was identified that in regular design space, the variations of aerodynamic and structural parameters are dominated by the airfoil camber and radius of leading edge. Meanwhile, the chord-wise position of the maximum thickness also has strong impacts on the airfoil performance. In further, the overall design spaces are explored to be highly nonlinear in aerodynamic and acoustic responses because of the nonlinear effects of the airfoil chord-wise position of the maximum camber and radius of leading edge. Strong but undesirable correlations were demonstrated between the maximum lift-to-drag ratio and the total sound pressure level. These findings could serve as a valuable guidance for wind turbine airfoil robust design to screen the stochastic design variables, simplify the design space, and reduce the cost.

Balanced caterpillars of maximum degree 3 and with hairs of arbitrary length are subgraphs of their optimal hypercube

AbstractA caterpillar is a tree having a path that contains all vertices of of degree at least 3. We show in this article that every balanced caterpillar with maximum degree 3 and 2n vertices is a subgraph of the n‐dimensional hypercube. This solves a long‐standing open problem and generalizes a result of Havel and Liebl (1986), who considered only such caterpillars that have a path containing all vertices of degree at least 2.

Embedding of special classes of circulant networks, hypercubes and generalized Petersen graphs

Hypercubes are a very popular model for parallel computation because of their regularity and the relatively small number of interprocessor connections. In this paper, we present an algorithm for embedding special class of circulant networks into their optimal hypercubes with dilation 2 and prove its correctness. Also, we embed special class of circulant networks into special class of generalized Petersen graphs with dilation 2 and vice versa.

An information theoretic approach to generating fuzzy hypercubes for if-then classifiers

In this paper, a framework for automatic generation of fuzzy membership functions and fuzzy rules from training data is proposed. The main focus of this paper is designing fuzzy if-then classifiers; however the proposed method can be employed in designing a wide range of fuzzy system applications. After the fuzzy membership functions are modeled by their supports, an optimization technique, based on a multi-objective real coded genetic algorithm with adaptive cross over and mutation probabilities, is implemented to find near optimal supports. Employing interpretability constraint in parameter representation and encoding, we ensure that the generated fuzzy membership function does have a semantic meaning. The fitness function of the genetic algorithm, which estimates the quality of the generated membership functions, consists of two elements: (i) the Shannon entropy and mutual information measures to measure diversity of the data distribution in a hypercube; and (ii) the number of generated fuzzy rules addressing the measure of compactness of the fuzzy system. Finally membership functions are tuned to yield optimal classifier hypercubes, which represent the predictivity and discriminating power of the classifier. Fuzzy rules of the classifier are derived from the optimal hypercubes. Using the proposed approach to designing fuzzy if-then classifiers, we are also able to evaluate the generated membership functions and compare the results with that of other techniques which have been previously reported in the literature.Using the experimental result, we show that the proposed approach outperforms other techniques in low resolutions. It means that theproposed approach can achieve satisfying result with lower complexity.

On Embedding of m-Sequential k-ary Trees into Hypercubes

In this paper, we present an algorithm for embedding an m-sequential k-ary tree into its optimal hypercube with dilation at most 2 and prove its correctness.

On embedding subclasses of height-balanced trees in hypercubes

A height-balanced tree is a rooted binary tree T in which for every vertex v ∈ V ( T ) , the heights of the subtrees, rooted at the left and right child of v, differ by at most one; this difference is called the balance factor of v. These trees are extensively used as data structures for sorting and searching. We embed several subclasses of height-balanced trees of height h in Q h + 1 under certain conditions. In particular, if a tree T is such that the balance factor of every vertex in the first three levels is arbitrary (0 or 1) and the balance factor of every other vertex is zero, then we prove that T is embeddable in its optimal hypercube with dilation 1 or 2 according to whether it is balanced or not.

Simultaneous model building and validation with uniform designs of experiments

This article describes an implementation of a particular design of experiment (DoE) plan based upon optimal Latin hypercubes that have certain space-filling and uniformity properties with the goal of maximizing the information gained. The feature emphasized here is the concept of simultaneous model building and model validation plans whose union contains the same properties as the component sets. Two Latin hypercube DoE are constructed simultaneously for use in a meta-modelling context for model building and model validation. The goal is to optimize the uniformity of both sets with respect to space-filling properties of the designs whilst satisfying the key concept that the merged DoE, comprising the union of build and validation sets, has similar space-filling properties. This represents a development of an optimal sampling approach for the first iteration—the initial model building and validation where most information is gained to take the full advantage of parallel computing. A permutation genetic algorithm using several genetic operator strategies is implemented in which fitness evaluation is based upon the Audze-Eglais potential energy function, and an example is presented based upon the well-known six-hump camel back function. The relative efficiency of the strategies and the associated computational aspects are discussed with respect to the quality of the designs obtained. The requirement for such design approaches arises from the need for multiple calls to traditionally expensive system and discipline analyses within iterative multi-disciplinary optimisation frameworks.

Optimal hypercube simulation on the partitioned optical passive stars network

Partitioned Optimal Passive Stars network, POPS(d,g), is an optical interconnection network of N processors (N=dg) with g 2 optical passive star couplers. In this network, there are g groups of d processors each and the g 2 couplers are used for connecting each group with each of the groups, including itself. In this paper, we present a technique for optimally simulating a frequently arising hypercube communication pattern on this network for all combinations of values of d and g. Specifically, we show that one-hop movements on the hypercube along the same dimension can be simulated on the POPS(d,g) network in $\lceil \frac{d}{g}\rceil$ slots for d?g and in 2 slots for d=g.

Dense sets and embedding binary trees into hypercubes

The purpose of this paper is to describe a method for embedding binary trees into hypercubes based on an iterative embedding into their subgraphs induced by dense sets. As a particular application, we present a class of binary trees, defined in terms of size of their subtrees, whose members allow a dilation two embedding into their optimal hypercubes. This provides a partial evidence in favor of a long-standing conjecture of Bhatt and Ipsen which claims that such an embedding exists for an arbitrary binary tree.

Embedding some hierarchical caterpillars into hypercube

Abstract In this paper, we give a dilation 2 embedding of some hierarchical caterpillars into their optimal hypercube.

Efficient Embeddings into Hypercube-like Topologies

Embeddings of various graph classes into hypercubes have been widely studied. Almost all these classes are regularly structured graphs such as meshes, complete trees or pyramids. In this paper, we present a general method for one-to-one embeddings of irregularly structured graphs into their optimal hypercubes, based on extended edge bisectors of graphs. An extended edge bisector is an edge bisector with the additional property that a certain subset of the vertices is distributed more or less evenly among the two halves of the bisected graph. The dilation and congestion of our embedding depends on the quality of the extended edge bisector. Moreover, if the extended bisection can be computed efficiently on the hypercube, then so can the embedding. Our embedding technique can also be applied to embeddings into hypercube-like topologies such as folded hypercubes, twisted cubes, crossed cubes, M¨ obius cubes, Fibonacci cubes or star graphs.

Embedding Graphs with Bounded Treewidth into Their Optimal Hypercubes

In this paper, we present a one-to-one embedding of a graph with bounded treewidth into its optimal hypercube. This is the first time that embeddings of graphs with a highly irregular structure into hypercubes are investigated. The presented embedding achieves dilation of at most 3⌈log((d+1)(t+1))⌉+8 and node-congestion of at most O(d(dt)3), where t denotes the treewidth of the graph and d denotes the maximal degree of a vertex in the graph. Provided that the graph is given by its tree-decomposition the embedding can be computed efficiently on the hypercube itself. In particular, the embedding of a graph with constant treewidth and constant degree can be computed in time O(log2(n)logloglog(n)log*(n)). For graphs with constant treewidth, a minimal tree-decomposition can be computed efficiently in parallel due to a result of Bodlaender and Hagerup. In this case, the embedding can be computed on the hypercube in time O(log2(n)(d2+log(n)loglog2(n))).

Sparse Hypercube 3-spanners

A t-spanner of a graph G=(V,E), is a sub-graph S G=(V,E′) , such that E′⊆E and for every edge {u,v}∈E, there is a path from u to v in S G of length at most t. A minimum-edge t-spanner of a graph G, S G′ , is the t-spanner of G with the fewest edges. For general graphs and for t=2, the problem of determining for a given integer s, whether |E(S G′)|⩽s is NP-Complete (Peleg and Schaffer, J. Graph Theory 13(1) (1989) 99–116). Peleg and Ullman (SIAM J. Comput. 18(4) (1989) 740–747), give a method for constructing a 3-spanner of the n-vertex Hypercube with fewer than 7n edges. In this paper we give an improved construction giving a 3-spanner of the n-vertex Hypercube with fewer than 4n edges and we present a lower bound of 3n/2− o(1) on the size of the optimal Hypercube 3-spanner.

Extracting and implementing list homomorphisms in parallel program development

Homomorphisms are functions that match the divide-and-conquer pattern and are widely used in parallel programming. Two problems are studied for homomorphisms on lists: 1. (1) parallelism extraction: finding a homomorphic representation of a given function; 2. (2) parallelism implementation: deriving an efficient parallel program that computes the function. The proposed approach to parallelism extraction starts by writing two sequential programs for the function, on traditional cons lists and on dual snoc lists; the parallel program is obtained by generalizing sequential programs as terms. For almost-homomorphic functions, e.g., the maximum segment sum problem, our method provides a systematic embedding into a homomorphism. The implementation problem is addressed by introducing the class of distributable homomorphisms and deriving for them a common parallel program schema. The derivation is based on equational reasoning in the Bird-Meertens formalism, which guarantees the correctness of the parallel target program. The approach is illustrated with the function scan (parallel prefix), for which the combination of our two systematic methods yields the optimal hypercube algorithm, usually presented ad hoc in the literature.

Embedding ofk-ary Complete Trees into Hypercubes with Uniform Load

The main result of this paper is an algorithm for embeddingk-ary complete trees into hypercubes with uniform load and asymptotically optimal dilation. The algorithm is fully scalable—the dimension of the hypercube can be chosen independent of the arity and height of the tree. The embedding enables to emulate optimally both Divide&Conquer computations onk-ary complete trees where only one level of nodes is active at a time and general computations based onk-ary complete trees where all tree nodes are active simultaneously. As a special case, we get an algorithm for embedding thek-ary complete tree of given height into its optimal hypercube with load 1 and asymptotically optimal dilation.

Embedding 2-Dimensional Grids Into Optimal Hypercubes with Edge-Congestion 1 or 2

This paper explores one-to-one embeddings of 2-dimensional grids into hypercubes. It is shown that each 2-dimensional grid can be embedded with edge-congestion 2 into its optimal hypercube (the smallest hypercube with at least as many nodes as the grid). Additionally, a technique is developed to embed many 2-dimensional grids into their optimal hypercubes with edge-congestion 1.

A New combinatorial approach to optimal embeddings of rectangles

An important problem in graph embeddings and parallel computing is to embed a rectangular grid into other graphs. We present a novel, general, combinatorial approach to (one-to-one) embedding rectangular grids into their ideal rectangular grids and optimal hypercubes. In contrast to earlier approaches of Aleliunas and Rosenberg, and Ellis (1982), our approach is based on a special kind of doubly stochastic matrix. We prove that any rectangular grid can be embedded into its ideal rectangular grid with dilation equal to the ceiling of the compression ratio, which is both optimal up to a multiplicative constant and a substantial generalization of previous work. We also show that any rectangular grid can be embedded into its nearly ideal square grid with dilation at most 3. Finally, we show that any rectangular grid can be embedded into its optimal hypercube with optimal dilation 2, a result previously obtained, after much research, through an ad hoc approach. Our results also imply optimal simulations of 2-dimensional mesh-connected parallel machines by hypercubes and mesh-connected machines, where each processor in the guest is simulated by exactly one processor in the host. >

A New Efficient Algorithm for Embedding an Arbitrary Binary Tree into Its Optimal Hypercube

Thed-dimensional binary hypercube is a very popular model of parallel computation. On the other hand, the execution of many algorithms can be represented by binary trees, making it desirable to simulate binary trees on a hypercube. In this paper, we present a simple one-to-one embedding of arbitrary binary trees into their optimal hypercubes with dilation 8 and constant node-congestion. The novelty of our method is the use of an intermediate quadtree data structure, which also permits the embedding to be efficiently computed on the hypercube itself.