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Z. Miller, D. Pritikin, I.H. Sudborough, "Near Embeddings of Hypercubes into Cayley Graphs on the Symmetric Group," IEEE Transactions on Computers, vol. 43, no. 1, pp. 1322, January, 1994.  
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@article{ 10.1109/12.250605, author = {Z. Miller and D. Pritikin and I.H. Sudborough}, title = {Near Embeddings of Hypercubes into Cayley Graphs on the Symmetric Group}, journal ={IEEE Transactions on Computers}, volume = {43}, number = {1}, issn = {00189340}, year = {1994}, pages = {1322}, doi = {http://doi.ieeecomputersociety.org/10.1109/12.250605}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
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TY  JOUR JO  IEEE Transactions on Computers TI  Near Embeddings of Hypercubes into Cayley Graphs on the Symmetric Group IS  1 SN  00189340 SP13 EP22 EPD  1322 A1  Z. Miller, A1  D. Pritikin, A1  I.H. Sudborough, PY  1994 KW  hypercube networks; graph theory; group theory; hypercubes; Cayley graphs; symmetric group; hypercube networks; star network; parallel computation; onetomany maps; interprocessor communication time; toggling bit i; onetoone embedding. VL  43 JA  IEEE Transactions on Computers ER   
Simulations of hypercube networks by certain Cayley graphs on the symmetric group are investigated. Let Q(k) be the familiar kdimensional hypercube, and let S(n) be the star network of dimension n defined as follows. The vertices of S(n) are the elements of the symmetric group of degree n, two vertices x and y being adjacent if xo(1,i)=y for some i. That is, xy is an edge if x and y are related by a transposition involving some fixed symbol (which we take to be /spl I.bold/1). This network has nice symmetry properties, and its degree and diameter are sublogarithmic as functions of the number of vertices, making it compare favorably with the hypercube network. These advantages of S(n) motivate the study of how well it can simulate other parallel computation networks, in particular, the hypercube. The first step in such a simulation is the construction of a onetoone map f:Q(k)/spl rarr/S(n) of dilation d, for d small. That is, one wants a map f such that images of adjacent points are at most distance d apart in S(n). An alternative approach, best applicable when onetoone maps are difficult or impossible to find, is the construction of a onetomany map g of dilation d, defined as follows. For each point x/spl isin/Q(k), there is an associated subset g(x)/spl sube/V(S(n)) such that for each edge xy in Q(k), every x'/spl isin/g(x) is at most distance d in S(n) from some y'/spl isin/g(y). Such onetomany maps allow one to achieve the low interprocessor communication time desired in the usual onetoone embedding underlying a simulation. This is done by capturing the local structure of Q(k) inside of S(n) (via the onetomany embedding) when the global structure cannot be so captured. Our results are the following. 1) There exist the following onetomany embeddings: a) f:Q(k)/spl rarr/S(3k+1) with dilation (f)=1; b) f:Q(11k+2)/spl rarr/S(13k+2) with dilation (f)=2. 2) There exists a onetoone embedding f:Q(n2/sup n/spl minus/1/)/spl rarr/S(2/sup n/) with dilation (f)=3.
[1] S. Akers, D. Harel, and B. Krishnamurthy, "The star graph: An attractive alternative to the ncube," inProc. Int. Conf. Parallel Processing, 1987, pp. 393400.
[2] S. Bhatt, F. Chung, J.W. Hong, F. T. Leighton, and A. L. Rosenberg, "Optimal simulations by Butterfly networks," inProc. 20th Symp. Theory Comput., May 1988, pp. 192204.
[3] B. Monien and I. H. Sudborough, "Simulating binary trees on hypercubes," inProc. AWOC Conf. VLSI Algorithms and Architectures, July 1988, Lecture Notes in Computer Science, vol. 319. Springer Verlag, 1988, pp. 170180.
[4] S. Akers and B. Krishnamurthy, "A grouptheoretic model for symmetric interconnection networks,"IEEE Trans. Comput., vol. C38, no. 4, pp. 555566, 1989.
[5] S. Akers and B. Krishnamurthy, "Group graphs as interconnection networks," inProc. 14th Int. Conf. Fault Tolerant Computing, 1984, pp. 422427.
[6] W. H. Gates and C. H. Papadimitriou, "Bounds for sorting by prefix reversal,"Discrete Math., vol. 27, pp. 4757, 1979.
[7] Z. Miller, D. Pritikin, and I. H. Sudborough, "Bounded dilation maps of hypercubes into Cayley graphs on the symmetric group," submitted.
[8] M. Fellows, "Encoding graphs in graphs," Ph.D. dissertation, Dept. of Comput. Sci., Univ. of California, San Diego, 1985.
[9] M. Nigam, S. Sahni, and B. Krishnamurthy, "Embedding hamiltonians and hypercubes in star interconnection graphs," inProc. Int. Conf. Parallel Processing, vol. 1990.