Embedding Star Networks into Hypercubes

• Authors: Saïd Bettayeb;Bin Cong;Mike Girou;I. Hal Sudborough
• Venue: IEEE Transactions on Computers
• Year: 1996

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 n-dimensional star network we exhibit: 1) a dialtion N驴 1 embedding of Sn into HN, where $N=\left\lceil {\log _2(n! )} \right\rceil $, 2) a dilation 2(d + 1) embedding of Sn into $H_{2d+n-1}$, where $d=\left\lceil {\log _2(\left\lceil {{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-\nulldelimiterspace} 2}} \right\rceil !)} \right\rceil $, 3) a dilation 2d + 2i embedding of $S_{2^im}$ into $H_{2^i\,d+i2^i\,m-2i+1}$, where $d=\left\lceil {\log _2(m\ !)} \right\rceil $, 4) a dilation L embedding of Sn into Hd, where $L=1+\left\lfloor {\log _2(n\ !)} \right\rfloor $, and d = (n驴 1)L, 5) a dilation $(k+1){{(k+2)} \mathord{\left/ {\vphantom {{(k+2)} 2}} \right. \kern-\nulldelimiterspace} 2}$ embedding of Sn into $H_{n(k+1)-2^{k+1}\,+1}$, where $k=\left\lfloor {\log _2(n-1)} \right\rfloor $, 6) a dilation 3 embedding of $S_{2k+1}$ into $H_{2k^2\,+k}$, and 7) a dilation 4 embedding of $S_{3k+2}$ into $H_{3k^2\,+3k+1}$.Some of the embeddings are, in fact, optimum, in both dilation and expansion for small values of n. We also show that the embedding of Sn into its optimum hypercube requires dilation $\Omega (\log _2n)$.