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On the k-ary hypercube

1995
*
Theoretical Computer Science
*

In this paper, we propose and analyze a new interconnection network,

doi:10.1016/0304-3975(94)00197-q
fatcat:cxx4qqzpwjcdrcpu3ffitbwj6e
*the**k*-*ary**hypercube*. This new architecture captures*the*advantages of*the*mesh network and those of*the*binary*hypercube*. ... It has a smaller degree than that of its equivalent binary*hypercube*(*the**one*with at least as many nodes) and has a smaller diameter than its equivalent mesh of processors. 0304-3975/95/%09.50 0 1995- ... Thus*the*desired Hamiltonian cycles of Qk(n+ 1) have been exhibited. 0 Dilation*one*embeddings in*k*-*ary**hypercubes**The*following lemma shows that*the*n-dimensional*k*-*ary**hypercube*contains grids whose ...##
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ON THE COMPUTATIONAL COMPLEXITY OF ROUTING IN FAULTY K-ARY N-CUBES AND HYPERCUBES

2012
*
Parallel Processing Letters
*

We equate a routing algorithm in a (faulty) interconnection network whose underlying graph is a

doi:10.1142/s012962641250003x
fatcat:qr45lgw4brdshgqz7awtuowsca
*k*-*ary*n-cube or a*hypercube*, that attempts to route a packet from a fixed source node to a fixed destination ... node, with*the*sub-digraph of (healthy) links potentially usable by this routing algorithm as it attempts to route*the*packet. ... Implicitly describing interconnection networks Consider*the**k*-*ary*n-cube Q*k*n or*the**hypercube*Q n . Everything we say below applies equally to*hypercubes*so we only focus here*on**k*-*ary*n-cubes. ...##
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On Embedding of m-Sequential k-ary Trees into Hypercubes

2010
*
Applied Mathematics
*

In this paper, we present an algorithm for embedding an m-sequential

doi:10.4236/am.2010.16065
fatcat:a5bkq4mj5zhz3cklk7ulc7lhjy
*k*-*ary*tree into its optimal*hypercube*with dilation at most 2 and prove its correctness. ... Eğecioğlu and Ibel [18] proved that*the*dynamic*k*-*ary*tree can be embedded into its asymptotic*hypercube*. Thus there is a vast literature*on*embedding trees into*hypercubes*. ... Theorem 2 An m-sequential*k*-*ary*tree can be embedded into its optimal*hypercube*with dilation at most 2. ...##
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Strong forms of orthogonality for sets of hypercubes

2012
*
Discrete Mathematics
*

For

doi:10.1016/j.disc.2012.03.008
fatcat:hvsur4pm4jdihit2oemebrrbfe
*hypercubes*of dimension d ≥ 2, we discuss several generalizations of*the*usual notion of pairwise orthogonality. ... We provide some constructions for complete sets of orthogonal*hypercubes*and also discuss connections to MDS codes. ... These results formed part of*the*Ph.D. dissertation of*the*first author. ...##
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A Comparatively Analysis and Performance of Logical Topologies of Embedded Hypercube (LTOEH) Interconnection Network

2015
*
International Journal of Computer Applications
*

A detailed analysis shows that

doi:10.5120/ijca2015907535
fatcat:p5ux4m62wfauhey6ri7oj5d2ca
*the*double loop*hypercube*DLH (m,d) network is compared with mesh and torus embedded*hypercube*, torus embedded*hypercube*a better interconnection network in*the*properties ... Also it has been proved with*the*computational results of entire system that*the*embedded*hypercube*interconnection networks build is highly scale up in terms of communication. ... In table 2 to reflects link complexity and torus, and*hypercube*lies in between ring and*k*-*ary*n-cube. ...##
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Fault tolerance measures for m-ary n-dimensional hypercubes based on forbidden faulty sets

1998
*
IEEE transactions on computers
*

Our study also shows that

doi:10.1109/12.707589
fatcat:uhs6ifwckjeszlghkmtihddiwu
*the*corresponding result based*on**the*traditional fault model (where*k*is zero) tends to underestimate network resilience of large networks such as m-*ary*n-dimensional*hypercubes*... In this paper, we study fault tolerance measures for m-*ary*n-dimensional*hypercubes*based*on**the*concept of forbidden faulty sets. ... Section 3 presents*the*result*on**the*minimum vertex cut set for a faulty m-*ary*n-dimensional*hypercube*in which each node has at least*k*healthy nodes. ...##
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Page 645 of IEEE Transactions on Computers Vol. 52, Issue 5
[page]

2003
*
IEEE Transactions on Computers
*

*The*proposed scheme is based

*on*isomorphic partitioning, which recur- sively partitions a

*k*-

*ary*n-cube into 2" number of £ -

*ary*n-cubes, where i represents

*the*ith partition step. ... However,

*the*processor allocation problem has not been adequately addressed for

*the*

*k*-

*ary*n-cube architecture, even though it has been studied extensively for

*hypercubes*and meshes. ...

##
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A Census of Small Latin Hypercubes

2008
*
SIAM Journal on Discrete Mathematics
*

*the*d-

*ary*loops). ... We also give an exact formula for

*the*number of (isomorphism classes of) d-

*ary*quasigroups of order 3 for every d. ... If

*the*

*hypercube*is regarded as

*the*table of values of a d-

*ary*quasigroup

*on*[n], then isomorphisms of

*the*

*hypercube*correspond to standard isomorphisms of

*the*quasigroup. ...

##
###
Distributed algorithms for building Hamiltonian cycles in k-ary n-cubes and hypercubes with faulty links

2006
*
12th International Conference on Parallel and Distributed Systems - (ICPADS'06)
*

*On*input:

*k*and n (specifying

*the*

*k*-

*ary*n-cube Q

*k*n ); F , a set of at most 2n − 2 faulty links; and v, a node of Q

*k*n ,

*the*algorithm outputs nodes v + and v − such that if Find-Ham-Cycle is executed ... We use our algorithms to obtain distributed algorithms to embed Hamiltonian cycles

*k*-

*ary*n-cubes and

*hypercubes*with faulty links; our

*hypercube*algorithm improves

*on*a recently-derived algorithm due to ...

*One*method of circumventing this problem, so as to still retain a "

*hypercube*-like" interconnection network, is to build parallel computers so that

*the*underlying topology is

*the*

*k*-

*ary*n-cube Q

*k*n . ...

##
###
On construction of orthogonal d-ary operations

2017
*
Publications de l'Institut Mathématique (Beograd)
*

We recall several connections between d-

doi:10.2298/pim1715109m
fatcat:puvos32xvves5ct32yhnureyom
*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. ... A d-*hypercube*of order n is an n × · · · × n d (d times) array with n d elements from a set Q of cardinality n. ... A set of j*k**hypercubes*of order n and dimension d is called mutually*k*-orthogonal if, given any*k**hypercubes*from*the*set, they are*k*-orthogonal. ...##
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Transversals in completely reducible multiary quasigroups and in multiary quasigroups of order 4
[article]

2017
*
arXiv
*
pre-print

An n-

arXiv:1612.01797v3
fatcat:3svmsgsgjrgszf3kpbxnvqo3ae
*ary*quasigroup f of order q is an n-*ary*operation over a set of cardinality q such that*the*Cayley table of*the*operation is an n-dimensional latin*hypercube*of order q. ... We find a lower bound*on**the*number of transversals for a vast class of completely reducible quasigroups. ...*The*author was supported by*the*Russian Science Foundation (grant 14-11-00555) (Sections 3-7) and*the*Moebius Contest Foundation for Young Scientists (Sections 1-2). ...##
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Applying traffic merging to datacenter networks

2012
*
Proceedings of the 3rd International Conference on Future Energy Systems Where Energy, Computing and Communication Meet - e-Energy '12
*

*The*problem of reducing energy usage in datacenter networks is an important

*one*. ... We examine

*the*idea of traffic merging using three different datacenter networks -flattened butterfly, mesh and

*hypercube*networks. ... A generic N ×

*K*merge (with

*K*≤ N ) is defined with

*the*property that if at most

*K*packets arrive

*on*

*the*N uplinks (i.e. from N links into

*the*switch) then

*the*

*K*packets are sent

*on*to

*K*sequential ports ...

##
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Embedding of tree networks into hypercubes

1985
*
Journal of Parallel and Distributed Computing
*

*The*

*hypercube*is a good host graph for

*the*embedding of networks of processors because of its low degree and low diameter. ... Graphs such as trees and arrays can be embedded into a

*hypercube*with small dilation and expansion costs, but there are classes of graphs which can be embedded into a

*hypercube*only with large expansion ... Thus a graph can be considered as a

*k*-

*ary*tree with additional edges joining nodes whose levels differ by at most

*one*. ...

##
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DISTRIBUTED ALGORITHMS FOR BUILDING HAMILTONIAN CYCLES IN k-ARY n-CUBES AND HYPERCUBES WITH FAULTY LINKS

2007
*
Journal of Interconnection Networks (JOIN)
*

., Q

doi:10.1142/s0219265907002016
fatcat:lm2uynm63jf6fjpjoudezw2mne
*k*−1*on*a*k*-*ary*(n − 1)-cube P to */ /* obtain*the*fault set G. ...*One*method of circumventing this problem, so as to still retain a "*hypercube*-like" interconnection network, is to build parallel computers so that*the*underlying topology is*the**k*-*ary*n-cube Q*k*n . ...##
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The priority broadcast scheme for dynamic broadcast in hypercubes and related networks

1999
*
Proceedings. Frontiers '99. Seventh Symposium on the Frontiers of Massively Parallel Computation
*

In particular,

doi:10.1109/fmpc.1999.750612
fatcat:yus7utagonc6zklf2zwvx5jhje
*the*algorithms for star graphs, generalized*hypercubes*, and*k*-*ary*n-cubes with*k*= O1 are also asymptotically optimal. ... We apply*the*routing scheme to other network topologies, including*k*-*ary*n-cubes, meshes, tori, star graphs, generalized*hypercubes*, as well as any symmetric network, for efficient dynamic broadcast. ... Introduction*Hypercubes*and*k*-*ary*n-cubes are among*the*most popular network topologies for parallel processing. Many commercial and experimental parallel computers are built based*on*these networks. ...
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