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On the k-ary hypercube

Saïd Bettayeb
1995 Theoretical Computer Science  
In this paper, we propose and analyze a new interconnection network, 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  ... 
doi:10.1016/0304-3975(94)00197-q fatcat:cxx4qqzpwjcdrcpu3ffitbwj6e

ON THE COMPUTATIONAL COMPLEXITY OF ROUTING IN FAULTY K-ARY N-CUBES AND HYPERCUBES

IAIN A. STEWART
2012 Parallel Processing Letters  
We equate a routing algorithm in a (faulty) interconnection network whose underlying graph is a 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.  ... 
doi:10.1142/s012962641250003x fatcat:qr45lgw4brdshgqz7awtuowsca

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

Indra Rajasingh, Bharati Rajan, Ramanathan Sundara Rajan
2010 Applied Mathematics  
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.  ...  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.  ... 
doi:10.4236/am.2010.16065 fatcat:a5bkq4mj5zhz3cklk7ulc7lhjy

Strong forms of orthogonality for sets of hypercubes

John T. Ethier, Gary L. Mullen
2012 Discrete Mathematics  
For 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.  ... 
doi:10.1016/j.disc.2012.03.008 fatcat:hvsur4pm4jdihit2oemebrrbfe

A Comparatively Analysis and Performance of Logical Topologies of Embedded Hypercube (LTOEH) Interconnection Network

Mohd. Kalamuddin, Mohd. Husain
2015 International Journal of Computer Applications  
A detailed analysis shows that 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.  ... 
doi:10.5120/ijca2015907535 fatcat:p5ux4m62wfauhey6ri7oj5d2ca

Fault tolerance measures for m-ary n-dimensional hypercubes based on forbidden faulty sets

Jie Wu, Guanghui Guo
1998 IEEE transactions on computers  
Our study also shows that 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.  ... 
doi:10.1109/12.707589 fatcat:uhs6ifwckjeszlghkmtihddiwu

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.  ... 

A Census of Small Latin Hypercubes

Brendan D. McKay, Ian M. Wanless
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.  ... 
doi:10.1137/070693874 fatcat:4zrdydkxdrgfrl44rnh6634fdq

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

I.A. Stewart
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 .  ... 
doi:10.1109/icpads.2006.35 dblp:conf/icpads/Stewart06 fatcat:jfmvn6l5avg4xlnuqay3r7lpsi

On construction of orthogonal d-ary operations

Smile Markovski, Aleksandra Mileva
2017 Publications de l'Institut Mathématique (Beograd)  
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.  ...  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.  ... 
doi:10.2298/pim1715109m fatcat:puvos32xvves5ct32yhnureyom

Transversals in completely reducible multiary quasigroups and in multiary quasigroups of order 4 [article]

Anna Taranenko
2017 arXiv   pre-print
An n-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).  ... 
arXiv:1612.01797v3 fatcat:3svmsgsgjrgszf3kpbxnvqo3ae

Applying traffic merging to datacenter networks

Alessandro Carrega, Suresh Singh, Raffaele Bolla, Roberto Bruschi
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  ... 
doi:10.1145/2208828.2208831 dblp:conf/eenergy/CarregaSBB12 fatcat:otjw4niiqjfn3cbrof6j7syksq

Embedding of tree networks into hypercubes

Angela Y Wu
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.  ... 
doi:10.1016/0743-7315(85)90026-7 fatcat:x6sbsgxkhbf6rblz3nknyqopcu

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

IAIN A. STEWART
2007 Journal of Interconnection Networks (JOIN)  
., Q 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 .  ... 
doi:10.1142/s0219265907002016 fatcat:lm2uynm63jf6fjpjoudezw2mne

The priority broadcast scheme for dynamic broadcast in hypercubes and related networks

Chi-Hsiang Yeh, E.A. Varvarigos, Hua Lee
1999 Proceedings. Frontiers '99. Seventh Symposium on the Frontiers of Massively Parallel Computation  
In particular, 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.  ... 
doi:10.1109/fmpc.1999.750612 fatcat:yus7utagonc6zklf2zwvx5jhje
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