Boolean N-cube Research Articles

Обзор конструкций и свойств кодов Грея

Cyclic enumeration of binary words of length n, where each pair of adjacent words differs in exactly one index, is called n-dimensional Gray code. Gray code determines hamiltonian cycle in boolean n-cube. In this paper we give a review of constructions and classifications of Gray codes. Constructions are divided into three main groups: recursive, toric and stream. We also give some of Gray code properties, as an example of applications of these constructions. In particular, we consider spectrum of edge directions, graphs of 2-subwords in transition sequence, local uniformity and others. Also some unresolved problems are given.

On the spectrum of Hamiltonian cycles in the n-cube

The spectrum of a Hamiltonian cycle (Gray code) in a Boolean n-cube is a sequence of n numbers, where the ith number is equal to the number of edges of the ith direction in the cycle. Necessary conditions for the existence of a Gray code with a given spectrum are known: all numbers are even and the sum of any k numbers is at least 2k, k=1,…,n. It is proved that for all dimensions n these necessary conditions are sufficient for the existence of a Gray code with the given spectrum.

Incomparable integrals and approximate calculation of monotone Boolean functions

The number of incomparable k-dimensional intervals in the Boolean n-cube is estimated. The result is used to estimate the complexity of approximate computation of an arbitrary monotone Boolean function of n variables.

On orientation-preserving automorphisms of Hamiltonian cycles in the N-dimensional Boolean cube

We obtain an upper bound for the order of the group of orientation-preserving automorphisms of a Hamiltonian cycle in the Boolean n-cube. We prove that the existence of a Hamiltonian cycle with the order of the group of orientation-preserving automorphisms attaining this upper bound is equivalent to the existence of a Hamiltonian cycle with an additional condition on the graph of orbits of a fixed automorphism of the n-cube.

Upper bounds on the queuenumber of k-ary n-cubes

A queue layout of a graph consists of a linear order of its vertices, and a partition of its edges into queues, such that no two edges in the same queue are nested. The minimum number of queues in a queue layout of a graph G, denoted by qn ( G ) , is called the queuenumber of G. Heath and Rosenberg [SIAM J. Comput. 21 (1992) 927–958] showed that boolean n-cube (i.e., the n-dimensional hypercube) can be laid out using at most n − 1 queues. Heath et al. [SIAM J. Discrete Math. 5 (1992) 398–412] showed that the ternary n-cube can be laid out using at most 2 n − 2 queues. Recently, Hasunuma and Hirota [Inform. Process. Lett. 104 (2007) 41–44] improved the upper bound on queuenumber to n − 2 for hypercubes. In this paper, we deal with the upper bound on queuenumber of a wider class of graphs called k-ary n-cubes, which contains hypercubes and ternary n-cubes as subclasses. Our result improves the previous bound in the case of ternary n-cubes. Let Q n k denote the n-dimensional k-ary cube. This paper contributes three main results as follows: (1) qn ( Q n 3 ) ⩽ 2 n − 3 if n ⩾ 3 . (2) qn ( Q n k ) ⩽ 2 n − 2 if n ⩾ 2 and 4 ⩽ k ⩽ 8 . (3) qn ( Q n k ) ⩽ 2 n − 1 if n ⩾ 1 and k ⩾ 9 .

On minimal coverings of the Boolean cube by centered antichains

We consider coverings of the Boolean n-cube B n ={0,1} n by families of its subsets of the special form which are called centered antichains. Each such subset consists of pairwise incomparable strings which have at least one common unit component; the collection of centered antichains also includes the one-element set containing the only string 0 ̃ n consisting entirely of zeros. It is established that for each n⩾1 the minimum number of centered antichains, the union of which covers the n-cube, equals n⌊ log 2 n⌋+2(n−2 ⌊ log 2 n⌋ )+2 . For each n, a minimal covering is constructed in an explicit form.

Logical Models of Molecular Shapes and Their Families

A new discrete mathematical model of molecular shape is proposed, making use of the partition property of a representation of molecular shape. According to its geometrical and topological structure, a molecular surface can be partitioned into unbounded two-dimensional subsets (domains) and some common subsets of closures of two or more domains. The sets of these domains as a base of a finite topology, containing the Boolean n-cube as a lower Boolean sub-lattice of this topology, defines the domain of the proposed logical model. A logical function can be obtained that reflects the properties of the topological domains as well as the interrelations on the set of domains. Based on classical or quantum-chemical representations of molecular shape, these models allow one the implementation of methods of logical diagnostics in chemistry, and the definition of a metric on the set of molecular shape equivalence classes. The families of molecular shapes can be considered as sets of logical models. The proposed model is unified in the sense that the structures of differentiable and non-differentiable surfaces can be represented in the same mathematical framework. These logical models will also work for interpenetrations of the above types of surfaces.

Parallel computer vision on a reconfigurable multiprocessor network

A novel reconfigurable architecture based on a multiring multiprocessor network is described. The reconfigurability of the architecture is shown to result in a low network diameter and also a low degree of connectivity for each node in the network. The mathematical properties of the network topology and the hardware for the reconfiguration switch are described. Primitive parallel operations on the network topology are described and analyzed. The architecture is shown to contain 2D mesh topologies of varying sizes and also a single one factor of the Boolean hypercube in any given configuration. A large class of algorithms for the 2D mesh and the Boolean n-cube are shown to map efficiently on the proposed architecture without loss of performance. The architecture is shown to be well suited for a number of problems in low and intermediate level computer vision such as the FFT, edge detection, template matching, and the Hough transform. Timing results for typical low and intermediate level vision algorithms on a transputer based prototype are presented.

The REFINE multiprocessor — theoretical properties and algorithms

A reconfigurable interconnection network based on a multi-ring architecture called REFINE is described. REFINE embeds a single 1-factor of the Boolean hypercube in any given configuration. The mathematical properties of the REFINE topology and the hardware for the reconfiguration switch are described. The REFINE topology is scalable in the sense that the number of interprocessor communication links scales linearly with network size whereas the network diameter scales logarithmically with network size. Primitive parallel operations on the REFINE topology are described and analyzed. These primitive operations could be used as building blocks for more complex parallel algorithms. A large class of algorithms for the Boolean n-cube which includes the FFT and the Batcher's bitonic sort is shown to map efficiently on the REFINE topology. The REFINE multiprocessor is shown to offer a cost-effective alternative to the Boolean n-cube multiprocessor architecture without substantial loss in performance.

Sphere packing numbers for subsets of the Boolean n-cube with bounded Vapnik-Chervonenkis dimension

Let V ⊆ {0, 1} n have Vapnik-Chervonenkis dimension d. Let M ( k/n, V) denote the cardinality of the largest W ⊆ V such that any two distinct vectors in W differ on at least k indices. We show that M ( k/n, V) ≤ ( cn/(k + d)) d for some constant c. This improves on the previous best result of (( cn k ) log( n k )) d . This new result has applications in the theory of empirical processes.

Modular fault-tolerant Boolean n-cubes

The modular design approach we propose for fault-tolerant Boolean n-cube architectures lets us reduce hardware cost and increase system reliability, while assuring high performance during message routing. Use of a specific module size with shared links and switches during reconfiguration helps keep system costs low. Though this strategy may increase the number of routing steps needed to transmit a message between nodes, routing performance nonetheless surpasses competing approaches, if we consider switching delays.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Improved lower bounds on the reliability of hypercube architectures

The hypercube topology, also known as the Boolean n-cube, has recently been used for multiprocessing systems. The paper considers two structural-reliability models, namely, terminal reliability (TR) and network reliability (NR), for the hypercube. Terminal (network) reliability is defined as the probability that there exists a working path connecting two (all) nodes. There are no known polynomial time algorithms for exact computation of TR or NR for the hypercube. Thus, lower-bound computation is a better alternative, because it is more efficient computationally, and the system will be at least as reliable as the bound. The paper presents algorithms to compute lower bounds on TR and NR for the hypercube considering node and/or link failures. These algorithms provide tighter bounds for both TR and NR than known results and run in time polynomial in the cube dimension n, specifically, within time O(n/sup 2/). >

A reconfigurable boolean n-cube architecture under faults

A reconfigurable modular Boolean n-cube (RMB n) is proposed in this paper. We embed spare elements, including nodes (processors), switches and links, into each module in order to reconfigure a failed system. The proposed scheme is constructed in two levels. The first level is to build a fault-tolerant module (FTM), using an original 4-cube of 16 nodes with 4 spare nodes and some switches with links. Then, in the second level, we use several FTMs to construct the desired RMB n, n ≥ 4, via connecting the corresponding nodes between any two adjacent FTMs. In our scheme, each spare node can replace any faulty node in the FTM via rebuilding the interconnections, i.e., each FTM can tolerate 4 faulty nodes. A reconfiguration algorithm is developed to choose an adequate spare node such that a faulty node can be replaced. We also develop a distributed routing algorithm to route messages from any source node to any destination node around the faulty nodes. Finally, the RMB n is evaluated and compared with previous works. It is shown that the proposed scheme can achieve the same or higher reliability while using fewer hardware costs.

Fault diagnosis for the generalized Boolean n-cube network

In this paper the fault-tolerant characteristics of the generalized Boolean n-cube network are studied. For processor failures the network is shown to be n+1 (one step) diagnosable. Two efficient algorithms which can diagnose up to n+1 processor failures are presented and their computational complexity are studied.

A reconfigurable modular fault tolerant generalized Boolean n-cube network

A fault tolerant modular architecture for generalized Boolean n-cube network is presented. In this approach, each module consists of 2 α original processors (nodes) and one spare processor (node), for 0≤ α≤n-1. The spare processor can replace each of the original processors within the same module. Thus, each module can tolerate one fault and the system can tolerante several faults if they are located in different modules. And then, a distributed fault tolerant routing algorithm is proposed. When a message destined to a faulty processor will be delivered to the spare that replaces the faulty processor. Finally, we show that the architecture can achieve very high reliability.

Graph theoretic characterization and reliability of the generalized Boolean n-cube network

In this paper, a general class of Boolean n-cube structures is investigated. The interconnection is based on a mixed radix number system which results in a variety of generalized Boolean n-cube structures for a given number of processors N = x·2 y , where x and y are positive integers. A number of interesting properties of the network are revealed. By a constructive method, the node connectivity of this network is found. Finally, we show that the graph is super-λ and is an optimal reliable structure for interconnection networks.

Spanning Balanced Trees in Boolean Cubes

A Spanning Balancedn-tree (SBnT) in a Boolean n-cube is a spanning tree in which the root has fanout n, and all the subtrees of the root have $O({{2^n } / n})$ nodes. The number of tree edges in each dimension of the n-cube is of order $O({{2^n } / n})$. The spanning balanced n-tree allows for scheduling disciplines that realize lower bound (within a factor of two) one-to-all personalized communication, all-to-all broadcasting, and all-to-all personalized communication on a Boolean n-cube [C.-T. Ho and S. L. Johnsson, Proc. 1986 International Conference on Parallel Processing, pp. 640–648, IEEE Computer Society, 1986; Tech. Report YALEU/DCS/RR–483, May 1986], [S. L. Johnsson and C.-T. Ho, Tech. Report YALEU/DCS/RR–610, Dept. of Computer Science, Yale Univ., New Haven, CT, November 1987]. The improvement in data transfer time over the familiar binomial tree routing is a factor of ${n / 2}$ for concurrent communication on all ports and one-to-all personalized communication and all-to-all broadcasting. For all-to-all personalized communication on all ports concurrently, the improvement is of order $O(\sqrt n )$. Distributed routing algorithms defining the spanning balanced n-tree are given. The balanced n-tree is not unique, and a few definitions of n-trees that are effectively edge-disjoint are provided. Some implementation issues are also discussed. Binary numbers obtained from each other through rotation form necklaces that are full if the period is equal to the length of the number; otherwise, they are degenerate. As an intermediary result, it is shown that the ratio between the number of degenerate necklaces and the total number of necklaces with l bits equal to one is at most ${4 / {(4 + n)}}$ for $1 \leqq l < n$.

Static interconnection network extensibility based on marginal performance/cost analysis

The paper examines the trade-off between cost and performance of a hypercube based computer interconnection network founded on a detailed cost analysis model. A cost function is proposed which considers the economies of scale of bandwidth available through optical fibre communication and the nonincremental increase in node complexity resulting from variations in network size. When considering topology choice, an important consideration is expansion. Any comparison of expansion should be done through marginal analysis: weighing the additional benefits (performance) against the additional cost resulting from expansion. To demonstrate this technique, four interconnection networks are considered: boolean n-cube, nearest neighbour mesh hypercube, generalised hypercube, and spanning multiaccess channel hypercube. Systems with equalised performance are studied, investigating the variations in cost. This permits the investigation of the trade-off of performance/cost between high capacity communication channels and increased number of channels and degree.

Optimum broadcasting and personalized communication in hypercubes

Four different communication problems are addressed in Boolean n-cube configured multiprocessors: (1) one-to-all broadcasting: distribution of common data from a single source to all other nodes; (2) one-to-all personalized communication: a single node sending unique data to all other nodes; (3) all-to-all broadcasting: distribution of common data from each node to all other nodes; and (4) all-to-all personalized communication: each node sending a unique piece of information to every other node. Three communication graphs (spanning trees) for the Boolean n-cube are proposed for the routing, and scheduling disciplines provably optimum within a small constant factor are proposed. With appropriate scheduling and concurrent communication on all ports of every processor, routings based on these two communication graphs offer a speedup of up to n/2, and O( square root n) over the routings based on the spanning binomial tree for cases (2)-(4) respectively. All three spanning trees offer optimal communication times for cases (2)-(4) and concurrent communication on all ports of every processor. Timing models and complexity analysis are verified by experiments on a Boolean-cube-configured multiprocessor. >

Algorithms for Matrix Transposition on Boolean N-Cube Configured Ensemble Architectures

In a multiprocessor with distributed storage the data structures have a significant impact on the communication complexity. In this paper we present a few algorithms for performing matrix transposition on a Boolean n-cube. One algorithm performs the transpose in a time proportional to the lower bound both with respect to communication start-ups and to element transfer times. We present algorithms for transposing a matrix embedded in the cube by a binary encoding, a binary-reflected Gray code encoding of rows and columns, or combinations of these two encodings. The transposition of a matrix when several matrix elements are identified to a node by consecutive or cyclic partitioning is also considered and lower bound algorithms given. Experimental data are provided for the Intel iPSC and the Connection Machine.