Issue No. 03 - March (1998 vol. 9)

ISSN: 1045-9219

pp: 261-274

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/71.674318

ABSTRACT

<p><b>Abstract</b>—We discuss the problem of packing hypercubes into an <it>n</it>-dimensional star graph <it>S</it>(<it>n</it>), which consists of embedding a disjoint union of hypercubes <it>U</it> into <it>S</it>(<it>n</it>) with load one. Hypercubes in <it>U</it> have from <tmath>$\lfloor n/2 \rfloor$</tmath> to <tmath>$(n+1)\cdot \left\lfloor {\log_2\,n} \right\rfloor -2^{\left\lfloor {\log_2n} \right \rfloor +1}+2$</tmath> dimensions, i.e., they can be as large as any hypercube which can be embedded with dilation at most four into <it>S</it>(<it>n</it>). We show that <it>U</it> can be embedded into <it>S</it>(<it>n</it>) with optimal expansion, which contrasts with the growing expansion ratios of previously known techniques.</p><p>We employ several performance metrics to show that, with our techniques, a star graph can efficiently execute heterogeneous workloads containing hypercube, mesh, and star graph algorithms. The characterization of our packings includes some important metrics which have not been addressed by previous research (namely, average dilation, average congestion, and congestion). Our packings consistently produce small average congestion and average dilation, which indicates that the induced communication slowdown is also small. We consider several combinations of node mapping functions and routing algorithms in <it>S</it>(<it>n</it>), and obtain their corresponding performance metrics using either mathematical analysis or computer simulation.</p>

INDEX TERMS

Graph embeddings, hypercubes, interconnection networks, routing, star graphs.

CITATION

Shahram Latifi, Nader Bagherzadeh, Marcelo Moraes de Azevedo, "Low Expansion Packings and Embeddings of Hypercubes into Star Graphs: A Performance-Oriented Approach",

*IEEE Transactions on Parallel & Distributed Systems*, vol. 9, no. , pp. 261-274, March 1998, doi:10.1109/71.674318