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Journal articles on the topic 'Hypercubes'

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Published: 4 June 2021

Last updated: 4 May 2024

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1

Röttger, Markus, and Ulf-Peter Schroeder. "Embedding 2-Dimensional Grids Into Optimal Hypercubes with Edge-Congestion 1 or 2." Parallel Processing Letters 08, no. 02 (June 1998): 231–42. http://dx.doi.org/10.1142/s0129626498000249.

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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.

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2

ZIAVRAS, SOTIRIOS G. "SCALABLE MULTIFOLDED HYPERCUBES FOR VERSATILE PARALLEL COMPUTERS." Parallel Processing Letters 05, no. 02 (June 1995): 241–50. http://dx.doi.org/10.1142/s0129626495000229.

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This paper introduces the family of scalable multifolded hypercube (SMH) architectures for parallel computers. Scalability and versatility at resonable cost are the major characteristics of these architectures. SMHs perform comparable to generalized hypercubes for important classes of algorithms that use regular communication patterns. In addition, they often achieve better performance than the popular direct binary hypercubes because they can emulate efficiently a powerful family of multifolded direct binary hypercubes. Extensive comparison of cost with binary and generalized hypercubes is also included. The hardware cost of SMH's is shown to be even lower than that of fat trees. Therefore, SMH's are viable candidates for the construction of versatile parallel computers.

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3

Burkov, Andriy, and Brahim Chaib-draa. "An Approximate Subgame-Perfect Equilibrium Computation Technique for Repeated Games." Proceedings of the AAAI Conference on Artificial Intelligence 24, no. 1 (July 4, 2010): 729–36. http://dx.doi.org/10.1609/aaai.v24i1.7623.

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This paper presents a technique for approximating, up to any precision, the set of subgame-perfect equilibria (SPE) in repeated games with discounting. The process starts with a single hypercube approximation of the set of SPE payoff profiles. Then the initial hypercube is gradually partitioned on to a set of smaller adjacent hypercubes, while those hypercubes that cannot contain any SPE point are gradually withdrawn. Whether a given hypercube can contain an equilibrium point is verified by an appropriate mixed integer program. A special attention is paid to the question of extracting players' strategies and their representability in form of finite automata.

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4

HUANG, KE, and JIE WU. "AREA EFFICIENT LAYOUT OF BALANCED HYPERCUBES." International Journal of High Speed Electronics and Systems 06, no. 04 (December 1995): 631–45. http://dx.doi.org/10.1142/s0129156495000237.

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As a multicomputer structure, the balanced hypercube is a variant of the standard hypercube for multicomputers, with desirable properties of strong connectivity, regularity, and symmetry. This structure is a special type of load balanced graph designed to tolerate processor failure. In balanced hypercubes, each processor has a backup (matching) processor that shares the same set of neighboring nodes. Therefore, tasks that run on a faulty processor can be reactivated in the backup processor to provide efficient system reconfiguration. In this paper, we study the implementation of balanced hypercubes in VLSI using the Wafer Scale Integration (VLSI/WSI) technology. Emphasis is on VLSI/WSI layout and area estimates. Our results show that the balanced hypercube can be implemented at least as efficient as the standard hypercube in an area layout and more efficient in a linear layout.

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5

Kim, Jin S., Seung Ryoul Maeng, and H. Yoon. "Ring Embedding in Hypercubes with Faculty Nodes." Parallel Processing Letters 07, no. 03 (September 1997): 285–96. http://dx.doi.org/10.1142/s0129626497000309.

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Hypercube is an attractive structure for parellel processing due to its symmetry and regularity. To increase the reliability of hypercube based systems and to allow their use in the presence of faulty nodes, efficient fault-tolerant schemes in hypercubes are necessary. In this paper, we present an algorithm for embedding rings in hypercubes based multiprocessor network in the event of node failures. The algorithm can tolerate up to θ(2n/2) faults, and guarantee that given any f < (n - 2k)2k faulty nodes, it can find a ring of size at least 2n - 2f for k = 0 and 2n - 2k f - 22k for k ≥ 1 in an n-dimensional hypercube. It improves over existing algorithms in the size of ring.

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6

Liu, Jia-Bao, Xiang-Feng Pan, and Jinde Cao. "Some Properties on Estrada Index of Folded Hypercubes Networks." Abstract and Applied Analysis 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/167623.

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LetGbe a simple graph withnvertices and letλ1,λ2,…,λnbe the eigenvalues of its adjacency matrix; the Estrada indexEEGof the graphGis defined as the sum of the termseλi, i=1,2,…,n. Then-dimensional folded hypercube networksFQnare an important and attractive variant of then-dimensional hypercube networksQn, which are obtained fromQnby adding an edge between any pair of vertices complementary edges. In this paper, we establish the explicit formulae for calculating the Estrada index of the folded hypercubes networksFQnby deducing the characteristic polynomial of the adjacency matrix in spectral graph theory. Moreover, some lower and upper bounds for the Estrada index of the folded hypercubes networksFQnare proposed.

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7

LATIFI, SHAHRAM. "SUBCUBE EMBEDDABILITY OF FOLDED HYPERCUBES." Parallel Processing Letters 01, no. 01 (September 1991): 43–50. http://dx.doi.org/10.1142/s0129626491000203.

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The Folded Hypercube (FHC) has been proven to be an attractive hypercube-based network. This paper closely compares the FHC to its standard hypercube counterpart from the subcube allocation viewpoint. It is shown that the FHC(n) outperforms the n-dimensional hypercube (n-cube for short) in offering subcubes of size k by a factor of [Formula: see text]. In an environment where subcubes of the original network must be allocated to incoming tasks, the FHC achieves an excellent processor utilization by assigning subcubes in an efficient and compact manner. Using the concept of virtual hypercubes, an efficient way is suggested to recognize the available subcubes in the FHC by adapting the already developed subcube recognition algorithms. An alternative approach to the subcube recognition problem is also given.

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8

Klavžar, Sandi. "Counting hypercubes in hypercubes." Discrete Mathematics 306, no. 22 (November 2006): 2964–67. http://dx.doi.org/10.1016/j.disc.2005.10.036.

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9

BEST, ANA, MARKUS KLIEGL, SHAWN MEAD-GLUCHACKI, and CHRISTINO TAMON. "MIXING OF QUANTUM WALKS ON GENERALIZED HYPERCUBES." International Journal of Quantum Information 06, no. 06 (December 2008): 1135–48. http://dx.doi.org/10.1142/s0219749908004377.

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We study continuous-time quantum walks on graphs which generalize the hypercube. The only known family of graphs whose quantum walk instantaneously mixes to uniform is the Hamming graphs with small arities. We show that quantum uniform mixing on the hypercube is robust under the addition of perfect matchings but not much else. Our specific results include: • The graph obtained by augmenting the hypercube with an additive matching x ↦ x ⊕ η is instantaneous uniform mixing whenever |η| is even, but with a slower mixing time. This strictly includes the result of Moore and Russell1 on the hypercube. • The class of Hamming graphs H(n,q) is not uniform mixing if and only if q ≥ 5. This is a tight characterization of quantum uniform mixing on Hamming graphs; previously, only the status of H(n,q) with q < 5 was known. • The bunkbed graph [Formula: see text] whose adjacency matrix is I ⊗ Qn + X ⊗ Af, where Af is a [Formula: see text]-circulant matrix defined by a Boolean function f, is not uniform mixing if the Fourier transform of f has support of size smaller than 2n-1. This explains why the hypercube is uniform mixing and why the join of two hypercubes is not. Our work exploits the rich spectral structure of the generalized hypercubes and relies heavily on Fourier analysis of group-circulants.

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10

TEL, GERARD. "LINEAR ELECTION IN HYPERCUBES." Parallel Processing Letters 05, no. 03 (September 1995): 357–66. http://dx.doi.org/10.1142/s0129626495000333.

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This article proposes an election algorithm for the hypercube; it exchanges less than [Formula: see text] messages and uses O( log 2 N) time (where N is the size of the cube). A randomized version of the algorithm achieves the same (expected) asymptotic message and time bounds, but uses messages of only O( log log N) bits and can be used in anonymous hypercubes.

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11

Kumar, J. M., and L. M. Patnaik. "Extended hypercube: a hierarchical interconnection network of hypercubes." IEEE Transactions on Parallel and Distributed Systems 3, no. 1 (1992): 45–57. http://dx.doi.org/10.1109/71.113081.

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12

Chang, Hsuan-Han, Kuan-Ting Chen, and Pao-Lien Lai. "How to Systematically Embed Cycles in Balanced Hypercubes." International Journal of Software Innovation 5, no. 1 (January 2017): 44–56. http://dx.doi.org/10.4018/ijsi.2017010104.

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The balanced hypercube is a variant of the hypercube structure and has desirable properties like connectivity, regularity, and symmetry. The cycle is a popular interconnection topology and has been widely used in distributed-memory parallel computers. Moreover, parallel algorithms of cycles have been extensively developed and used. The problem of how to embed cycles into a host graph has attracted a great attention in recent years. However, there is no systematic method proposed to generate the desired cycles in balanced hypercubes. In this paper, the authors develop systematic linear time algorithm to construct cycles and Hamiltonian cycles for the balanced hypercube.

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José, Marco V., Eberto R. Morgado, and Juan R. Bobadilla. "Groups of Symmetries of the Two Classes of Synthetases in the Four-Dimensional Hypercubes of the Extended Code Type II." Life 13, no. 10 (September 30, 2023): 2002. http://dx.doi.org/10.3390/life13102002.

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Aminoacyl-tRNA synthetases (aaRSs) originated from an ancestral bidirectional gene (mirror symmetry), and through the evolution of the genetic code, the twenty aaRSs exhibit a symmetrical distribution in a 6-dimensional hypercube of the Standard Genetic Code. In this work, we assume a primeval RNY code and the Extended Genetic RNA code type II, which includes codons of the types YNY, YNR, and RNR. Each of the four subsets of codons can be represented in a 4-dimensional hypercube. Altogether, these 4 subcodes constitute the 6-dimensional representation of the SGC. We identify the aaRSs symmetry groups in each of these hypercubes. We show that each of the four hypercubes contains the following sets of symmetries for the two known Classes of synthetases: RNY: dihedral group of order 4; YNY: binary group; YNR: amplified octahedral group; and RNR: binary group. We demonstrate that for each hypercube, the group of symmetries in Class 1 is the same as the group of symmetries in Class 2. The biological implications of these findings are discussed.

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14

ZIAVRAS, SOTIRIOS G., and MICHALIS A. SIDERAS. "FACILITATING HIGH-PERFORMANCE IMAGE ANALYSIS ON REDUCED HYPERCUBE (RH) PARALLEL COMPUTERS." International Journal of Pattern Recognition and Artificial Intelligence 09, no. 04 (August 1995): 679–98. http://dx.doi.org/10.1142/s0218001495000262.

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The direct binary hypercube interconnection network has been very popular for the design of parallel computers, because it provides a low diameter and can emulate efficiently the majority of the topologies frequently employed in the development of algorithms. The last fifteen years have seen major efforts to develop image analysis algorithms for hypercube-based parallel computers. The results of these efforts have culminated in a large number of publications included in prestigious scholarly journals and conference proceedings. Nevertheless, the aforementioned powerful properties of the hypercube come at the cost of high VLSI complexity due to the increase in the number of communication ports and channels per PE (processing element) with an increase in the total number of PE’s. The high VLSI complexity of hypercube systems is undoubtedly their dominant drawback; it results in the construction of systems that contain either a large number of primitive PE’s or a small number of powerful PE’s. Therefore, low-dimensional k-ary n-cubes with lower VSLI complexity have recently drawn the attention of many designers of parallel computers. Alternative solutions reduce the hypercube’s VLSI complexity without jeopardizing its performance. Such an effort by Ziavras has resulted in the introduction of reduced hypercubes (RH’s). Taking advantage of existing high-performance routing techniques, such as wormhole routing, an RH is obtained by a uniform reduction in the number of edges for each hypercube node. An RH can also be viewed as several connected copies of the well-known cube-connected-cycles network. The objective here is to prove that parallel computers comprising RH interconnection networks are definitely good choices for all levels of image analysis. Since the exact requirements of high-level image analysis are difficult to identify, but it is believed that versatile interconnection networks, such as the hypercube, are suitable for relevant tasks, we investigate the problem of emulating hypercubes on RH’s. The ring (or linear array), the torus (or mesh), and the binary tree are the most frequently used topologies for the development of algorithms in low-level and intermediate-level image analysis. Thus, to prove the viability of the RH for the two lower levels of image analysis, we introduce techniques for embedding the aforementioned three topologies into RH’s. The results prove the suitability of RH’s for all levels of image analysis.

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15

Zelina, Ioana, Mara Hajdu-Măcelaru, and Cristina Ţicală. "About the cube polynomial of Extended Fibonacci Cubes." Creative Mathematics and Informatics 27, no. 1 (2018): 95–100. http://dx.doi.org/10.37193/cmi.2018.01.13.

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The hypercube is one of the best model for the network topology of a distributed system. In this paper we determine the cube polynomial of Extended Fibonacci Cubes, which is the counting polynomial for the number of induced k-dimensional hypercubes in Extended Fibonacci Cubes.

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16

Lobov, Alexander A., and Mikhail B. Abrosimov. "About uniqueness of the minimal 1-edge extension of hypercube Q4." Prikladnaya Diskretnaya Matematika, no. 58 (2023): 84–93. http://dx.doi.org/10.17223/20710410/58/8.

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One of the important properties of reliable computing systems is their fault tolerance. To study fault tolerance, you can use the apparatus of graph theory. Minimal edge extensions of a graph are considered, which are a model for studying the failure of links in a computing system. A graph G* = (V*,α*) with n vertices is called a minimal k-edge extension of an n-vertex graph G = (V, α) if the graph G is embedded in every graph obtained from G* by deleting any of its k edges and has the minimum possible number of edges. The hypercube Qn is a regular 2n-vertex graph of order n, which is the Cartesian product of n complete 2-vertex graphs K2. The hypercube is a common topology for building computing systems. Previously, a family of graphs Q*n was described, whose representatives for n>1 are minimal edge 1-extensions of the corresponding hypercubes. In this paper, we obtain an analytical proof of the uniqueness of minimal edge 1-extensions of hypercubes for n≤4 and establish a general property of an arbitrary minimal edge 1-extension of a hypercube Qn for n>2: it does not contain edges connecting vertices, the distance between which in the hypercube is equal to 2.

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17

Guo, Litao, and Xiaofeng Guo. "Fault tolerance of hypercubes and folded hypercubes." Journal of Supercomputing 68, no. 3 (December 31, 2013): 1235–40. http://dx.doi.org/10.1007/s11227-013-1078-5.

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18

Suykens, J. A. K., and L. O. Chua. "n-Double Scroll Hypercubes in 1-D CNNs." International Journal of Bifurcation and Chaos 07, no. 08 (August 1997): 1873–85. http://dx.doi.org/10.1142/s021812749700145x.

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Unidirectional and diffusive coupling of identical n-double scroll cells in a one-dimensional cellular neural network is studied. Weak coupling between the cells leads to hyperchaos, with n-double scroll hypercube attractors observed in the common state subspace of the cells. Individually the cells remain behaving as n-double scrolls. The n-double scroll hypercubes are filled with multiple scrolls. Their birth goes through the mechanism of intermittency.

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19

BAL, DEEPAK, ANTHONY BONATO, WILLIAM B. KINNERSLEY, and PAWEŁ PRAŁAT. "Lazy Cops and Robbers on Hypercubes." Combinatorics, Probability and Computing 24, no. 6 (January 29, 2015): 829–37. http://dx.doi.org/10.1017/s0963548314000807.

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We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. We investigate the analogue of the cop number for this game, which we call the lazy cop number. Lazy Cops and Robbers was recently introduced by Offner and Ojakian, who provided asymptotic upper and lower bounds on the lazy cop number of the hypercube. By coupling the probabilistic method with a potential function argument, we improve on the existing lower bounds for the lazy cop number of hypercubes.

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20

Chen, Liang, Caiming Zhong, and Zehua Zhang. "Explanation of clustering result based on multi-objective optimization." PLOS ONE 18, no. 10 (October 27, 2023): e0292960. http://dx.doi.org/10.1371/journal.pone.0292960.

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Clustering is an unsupervised machine learning technique whose goal is to cluster unlabeled data. But traditional clustering methods only output a set of results and do not provide any explanations of the results. Although in the literature a number of methods based on decision tree have been proposed to explain the clustering results, most of them have some disadvantages, such as too many branches and too deep leaves, which lead to complex explanations and make it difficult for users to understand. In this paper, a hypercube overlay model based on multi-objective optimization is proposed to achieve succinct explanations of clustering results. The model designs two objective functions based on the number of hypercubes and the compactness of instances and then uses multi-objective optimization to find a set of nondominated solutions. Finally, an Utopia point is defined to determine the most suitable solution, in which each cluster can be covered by as few hypercubes as possible. Based on these hypercubes, an explanations of each cluster is provided. Upon verification on synthetic and real datasets respectively, it shows that the model can provide a concise and understandable explanations to users.

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21

WU, JIE. "TIGHT BOUNDS ON THE NUMBER OF l-NODES IN A FAULTY HYPERCUBE." Parallel Processing Letters 05, no. 02 (June 1995): 321–28. http://dx.doi.org/10.1142/s0129626495000308.

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The spanning binomial tree is one of most frequently used spanning tree structures to implement various parallel applications in multiprocessor systems, such as hypercubes. In this paper, we define an l-node as a root node of an incomplete spanning binomial tree of a hypercube, which is defined as a connected subtree of a spanning binomial tree with the same root node that connects all the nonfaulty nodes in the hypercube. We show that in an n-dimensional hypercube with m faulty nodes there are at least 2n − 2ml-nodes. This implies that at least half of the nodes of the hypercube are l-nodes if the number of faulty nodes is less than the dimension of the hypercube.

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22

LAVAULT, CHRISTIAN. "EMBEDDINGS INTO THE PANCAKE INTERCONNECTION NETWORK." Parallel Processing Letters 12, no. 03n04 (September 2002): 297–310. http://dx.doi.org/10.1142/s0129626402001002.

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Owing to its nice properties, the pancake is one of the Cayley graphs that were proposed as alternatives to the hypercube for interconnecting processors in parallel computers. In this paper, we present embeddings of rings, grids and hypercubes into the pancake with constant dilation and congestion. We also extend the results to similar efficient embeddings into star graph.

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23

Pike, David A. "Decycling Hypercubes." Graphs and Combinatorics 19, no. 4 (November 1, 2003): 547–50. http://dx.doi.org/10.1007/s00373-003-0529-9.

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24

Katseff, H. P. "Incomplete hypercubes." IEEE Transactions on Computers 37, no. 5 (May 1988): 604–8. http://dx.doi.org/10.1109/12.4611.

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25

Tzeng, N. F., and S. Wei. "Enhanced hypercubes." IEEE Transactions on Computers 40, no. 3 (March 1991): 284–94. http://dx.doi.org/10.1109/12.76405.

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26

Iványi, Antal, and János Madarász. "Perfect hypercubes." Electronic Notes in Discrete Mathematics 38 (December 2011): 475–80. http://dx.doi.org/10.1016/j.endm.2011.09.077.

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27

Fraigniaud, Pierre, Jean-Claude König, and Emmanuel Lazard. "Oriented hypercubes." Networks 39, no. 2 (January 29, 2002): 98–106. http://dx.doi.org/10.1002/net.10012.

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28

BOSSARD, ANTOINE. "ON THE DECYCLING PROBLEM IN HIERARCHICAL HYPERCUBES." Journal of Interconnection Networks 14, no. 02 (June 2013): 1350006. http://dx.doi.org/10.1142/s0219265913500060.

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Due to the huge number of CPU nodes involved in modern supercomputers, efficient CPU connection is challenging, and legacy simple network topologies such as hypercubes are no more suitable for physical reasons. The hierarchical hypercube (HHC) has been designed as a topology for interconnection network of massively parallel systems. An HHC is effectively able to link many nodes while retaining a low degree and a small diameter compared to a hypercube of the same size. In this paper, we address a fundamental problem inside an HHC, the decycling problem, which consists of finding a set of nodes as small as possible such that excluding these nodes from the network ensures a cycle-free topology. This problem has many important applications such as lock-free resource allocation and concurrent access. So, we propose in this paper an efficient algorithm finding in an HHC a decycling set of competitively small size.

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Sabir, Eminjan, and Jixiang Meng. "Structure fault tolerance of hypercubes and folded hypercubes." Theoretical Computer Science 711 (February 2018): 44–55. http://dx.doi.org/10.1016/j.tcs.2017.10.032.

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Padmavathi, SV, and K. Krishnan. "Weak Convex Domination in Hypercubes." Shanlax International Journal of Arts, Science and Humanities 8, S1-May (May 15, 2021): 50–53. http://dx.doi.org/10.34293/sijash.v8is1-may.4507.

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The n-cube Qn is the graph whose vertex set is the set of all n-dimensional Boolean vectors, two vertices being joined if and only if they differ in exactly one coordinate. The n-star graph Sn is a simple graph whose vertex set is the set of all n! permutations of {1, 2, · · · , n} and two vertices α and β are adjacent if and only if α(1)≠β(1) and α(i) ≠β(i) for exactly one i, i≠ 1. In this paper we determine weak convex domination number for hypercubes. Also convex, weak convex, m - convex and l1-convex numbers of star and hypercube graphs are determined.

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Das, Sajal K., Sabine Öhring, and Amit K. Banerjee. "Embeddings into Hyper Petersen Networks: Yet Another Hypercube-Like Interconnection Topology." VLSI Design 2, no. 4 (January 1, 1995): 335–51. http://dx.doi.org/10.1155/1995/95759.

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A new hypercube-like topology, called the hyper Petersen (HP) network, is proposed and analyzed, which is constructed from the well-known cartesian product of the binary hypercube and the Petersen graph of ten nodes.This topology is an attractive candidate for multiprocessor interconnection having such desirable properties as regularity, high symmetry and connectivity, and logarithmic diameter. For example, an n-dimensional hyper Petersen network, HPn, with N=1.25 * 2n nodes is a regular graph of degree and node-connectivity n and diameter n–1 , whereas an (n–1)-dimensional binary hypercube, Qn−1 , with the same diameter covers only 2n−1 nodes, each of degree (n–1). Thus the HP topology accommodates 2.5 times extra nodes than Qn−1 at the cost of increasing the node-degree by one. With the same degree and connectivity of n, the diameter of the HPn network is one less than that of Qn, yet having 1.25 times larger number of nodes.Efficient routing and broadcasting schemes are presented, and node-disjoint paths in HPn, are computed even under faulty conditions. The versatility of the hyper Petersen networks is emphasized by embedding rings, meshes, hypercubes and several tree-related topologies into it. Contrary to the hypercubes, rings of odd lengths, and a complete binary tree of height n–1 permit subgraph embeddings in HPn.

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LI, QIULI, and WANTAO NING. "Matching Preclusion for Exchanged Hypercubes." Journal of Interconnection Networks 19, no. 03 (September 2019): 1940008. http://dx.doi.org/10.1142/s0219265919400085.

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As spanning subgraphs of hypercubes, exchanged hypercubes contain less edges but maintain lots of desired properties of hypercubes. This paper considers matching preclusion, a kind of measures of edge-fault tolerance, of exchanged hypercubes EH(s, t). We show that EH(s, t) is maximally matched, that is, for s ≥ t, mp(EH(s, t)) = t + 1 and EH(s, t) is super matched if and only if (s, t) ≠ (1, 1). Comparing with results of matching preclusion for hypercubes, we conclude that exchanged hypercubes maintain the matching preclusion property of hypercubes, except for EH(1, 1).

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PLATA, OSCAR, TOMAS F. PENA, FRANCISCO F. RIVERA, and EMILIO L. ZAPATA. "AN EFFICIENT PROCESSOR ALLOCATION FOR NESTED PARALLEL LOOPS ON DISTRIBUTED MEMORY HYPERCUBES." Parallel Processing Letters 03, no. 02 (June 1993): 179–87. http://dx.doi.org/10.1142/s0129626493000228.

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We consider the static processor allocation problem for arbitrarily nested parallel loops on distributed memory, message-passing hypercubes. We present HYPAL (HYpercube Partitioning ALgorithm) as an efficient algorithm to solve this problem. HYPAL calculates an optimal set of partitions of the dimension of the hypercube, and assigns them to the set of iterations of the nested loop. Some considerations about the influence of the communication overhead in order to get a more realistic approach are considered. The main problem at this point is to obtain the communication pattern associated to the parallel program because it depends on scheduling and data distribution.

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Cho, Yunhi, and Seonhwa Kim. "Volume of Hypercubes Clipped by Hyperplanes and Combinatorial Identities." Electronic Journal of Linear Algebra 36, no. 36 (April 29, 2020): 228–55. http://dx.doi.org/10.13001/ela.2020.5085.

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There is an elegant expression for the volume of hypercube $[0,1]^n$ clipped by a single hyperplane. In the article, the formula is generalized to the case of more than one hyperplane. An important foundation for the result is Lawrence's formula and a way to weaken two restrictions of simplicity and non-parallelness in his formula is also considered. Several concrete volume formulas of clipped hypercubes are derived explicitly and the corresponding combinatorial identities are obtained as an application.

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35

XIANG, DONG, AI CHEN, and JIE WU. "LOCAL-SAFETY-INFORMATION-BASED BROADCASTING IN HYPERCUBE MULTICOMPUTERS WITH NODE AND LINK FAULTS." Journal of Interconnection Networks 02, no. 03 (September 2001): 365–78. http://dx.doi.org/10.1142/s0219265901000440.

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This paper presents a method to cope with fault-tolerant broadcasting in hypercube multicomputers with both node and link faults. The local safety concept is extended to faulty hypercubes with both node and link faults. The local-safety-based algorithm is used in a fully unsafe hypercube, where there is no safe node. A fully unsafe hypercube can be split into a set of maximal safe subcubes. We show that if these maximal safe subcubes meet certain requirements given in the paper, broadcasting can still be carried out successfully and in some case optimal broadcast is still possible. The method is extended to fault-tolerant routing and multicasting when the system contains both node and link faults.

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36

HSU, GUO-HUANG, CHIEH-FENG CHIANG, and JIMMY J. M. TAN. "COMPARISON-BASED CONDITIONAL DIAGNOSABILITY ON THE CLASS OF HYPERCUBE-LIKE NETWORKS." Journal of Interconnection Networks 11, no. 03n04 (September 2010): 143–56. http://dx.doi.org/10.1142/s0219265910002775.

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For system diagnosis, Lai et al.16 introduced a new measurement, called the conditional diagnosability, by adding a condition that no faulty set contains all the neighbors of any vertex in a network. Taking the hypercube as the target, Lai et al.16 (respectively, Hsu et al.13) estimated the PMC-based19 (respectively, the comparison-based18) conditional diagnosability as about four (respectively, three) times larger than the original diagnosability. In this paper, we extend the concept of conditional diagnosability to the generalized version of hypercubes, the class of hypercube-like networks. We prove that the conditional diagnosability of an n-dimensional hypercube-like network HLn is 3n - 5 under the comparison diagnosis model, for n ≥ 5.

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37

Markovski, Smile, and Aleksandra Mileva. "On construction of orthogonal d-ary operations." Publications de l'Institut Math?matique (Belgrade) 101, no. 115 (2017): 109–19. http://dx.doi.org/10.2298/pim1715109m.

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A d-hypercube of order n is an n x ... x nd (d times) array with nd elements from a set Q of cardinality n. We recall several connections between d-hypercubes of order n and d-ary operations of order n. We give constructions of orthogonal d-ary operations that generalize a result of Belyavskaya and Mullen. Our main result is a general construction of d-orthogonal d-ary operations from d-ary quasigroups.

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38

Lord, Nick. "72.11 Subdividing Hypercubes." Mathematical Gazette 72, no. 459 (March 1988): 47. http://dx.doi.org/10.2307/3617994.

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Chung, Fan R. K. "Pebbling in Hypercubes." SIAM Journal on Discrete Mathematics 2, no. 4 (November 1989): 467–72. http://dx.doi.org/10.1137/0402041.

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Livingston, Marilynn, and Quentin F. Stout. "Embeddings in hypercubes." Mathematical and Computer Modelling 11 (1988): 222–27. http://dx.doi.org/10.1016/0895-7177(88)90486-4.

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41

Glynn, David G. "Nonfactorizable nonsingular hypercubes." Designs, Codes and Cryptography 68, no. 1-3 (November 19, 2011): 195–203. http://dx.doi.org/10.1007/s10623-011-9585-y.

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42

Yang, Weihua, and Jixiang Meng. "Extraconnectivity of hypercubes." Applied Mathematics Letters 22, no. 6 (June 2009): 887–91. http://dx.doi.org/10.1016/j.aml.2008.07.016.

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Dudek, Andrzej, Xavier Pérez-Giménez, Paweł Prałat, Hao Qi, Douglas West, and Xuding Zhu. "Randomly twisted hypercubes." European Journal of Combinatorics 70 (May 2018): 364–73. http://dx.doi.org/10.1016/j.ejc.2018.01.013.

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Bai, Zhi-Dong, Luc Devroye, Hsien-Kuei Hwang, and Tsung-Hsi Tsai. "Maxima in hypercubes." Random Structures and Algorithms 27, no. 3 (2005): 290–309. http://dx.doi.org/10.1002/rsa.20053.

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Zhang, Yan-Qing, and Yi Pan. "Incomplete crossed hypercubes." Journal of Supercomputing 49, no. 3 (September 27, 2008): 318–33. http://dx.doi.org/10.1007/s11227-008-0239-4.

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Lin, Shang-wei, and Na-qi Fan. "Restricted Arc Connectivity of Unidirectional Hypercubes and Unidirectional Folded Hypercubes." Taiwanese Journal of Mathematics 23, no. 3 (June 2019): 529–43. http://dx.doi.org/10.11650/tjm/180808.

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Liu, Aixia, Shiying Wang, Jun Yuan, and Jing Li. "On g -extra conditional diagnosability of hypercubes and folded hypercubes." Theoretical Computer Science 704 (December 2017): 62–73. http://dx.doi.org/10.1016/j.tcs.2017.09.030.

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Al-Ezeh, Hasan, Omar A. AbuGheim, and Eman A. AbuHijleh. "Characterizing which powers of hypercubes and folded hypercubes are divisor graphs." Discussiones Mathematicae Graph Theory 35, no. 2 (2015): 301. http://dx.doi.org/10.7151/dmgt.1801.

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Arockiaraj, Micheal, Jasintha Quadras, Indra Rajasingh, and Arul Jeya Shalini. "Embedding hypercubes and folded hypercubes onto Cartesian product of certain trees." Discrete Optimization 17 (August 2015): 1–13. http://dx.doi.org/10.1016/j.disopt.2015.03.001.

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Qiao, Yalin, and Weihua Yang. "Edge disjoint paths in hypercubes and folded hypercubes with conditional faults." Applied Mathematics and Computation 294 (February 2017): 96–101. http://dx.doi.org/10.1016/j.amc.2016.09.002.

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