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<p><b>Abstract</b>—The star interconnection network has recently been suggested as an alternative to the hypercube. As hypercubes are often viewed as universal and capable of simulating other architectures efficiently, we investigate embeddings of star network into hypercubes. Ourt embeddings exhibit a marked trade-off between dilation and expansion. For the <it>n</it>-dimensional star network we exhibit: 1) a dialtion <it>N</it>− 1 embedding of <it>S</it><sub><it>n</it></sub> into <it>H</it><sub><it>N</it></sub>, where <tmath>$N=\left\lceil {\log _2(n! )} \right\rceil $,</tmath> 2) a dilation 2(<it>d</it> + 1) embedding of <it>S</it><sub><it>n</it></sub> into <tmath>$H_{2d+n-1}$,</tmath> where <tmath>$d=\left\lceil {\log _2(\left\lceil {{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} \right\rceil !)} \right\rceil $,</tmath> 3) a dilation 2<it>d</it> + 2<it>i</it> embedding of <tmath>$S_{2^im}$</tmath> into <tmath>$H_{2^i\,d+i2^i\,m-2i+1}$,</tmath> where <tmath>$d=\left\lceil {\log _2(m\ !)} \right\rceil $,</tmath> 4) a dilation <it>L</it> embedding of <it>S</it><sub><it>n</it></sub> into <it>H</it><sub><it>d</it></sub>, where <tmath>$L=1+\left\lfloor {\log _2(n\ !)} \right\rfloor $,</tmath> and <it>d</it> = (<it>n</it>− 1)<it>L</it>, 5) a dilation <tmath>$(k+1){{(k+2)} \mathord{\left/ {\vphantom {{(k+2)} 2}} \right. \kern-\nulldelimiterspace} 2}$</tmath> embedding of <it>S</it><sub><it>n</it></sub> into <tmath>$H_{n(k+1)-2^{k+1}\,+1}$,</tmath> where <tmath>$k=\left\lfloor {\log _2(n-1)} \right\rfloor $,</tmath> 6) a dilation 3 embedding of <tmath>$S_{2k+1}$</tmath> into <tmath>$H_{2k^2\,+k}$,</tmath> and 7) a dilation 4 embedding of <tmath>$S_{3k+2}$</tmath> into <tmath>$H_{3k^2\,+3k+1}$.</tmath></p><p>Some of the embeddings are, in fact, optimum, in both dilation and expansion for small values of <it>n</it>. We also show that the embedding of <it>S</it><sub><it>n</it></sub> into its optimum hypercube requires dilation <tmath>$\Omega (\log _2n)$.</tmath></p>
Hypercube, star network, embedding, dilation, expansion, permutation, Cayley graph.
Saïd Bettayeb, Mike Girou, Bin Cong, I. Hal Sudborough, "Embedding Star Networks into Hypercubes", IEEE Transactions on Computers, vol. 45, no. , pp. 186-194, February 1996, doi:10.1109/12.485371
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