Issue No. 08 - August (1995 vol. 44)

ISSN: 0018-9340

pp: 1021-1030

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/12.403718

ABSTRACT

<p><it>Abstract</it>—In this paper, we consider various topological properties of a <it>k</it>-ary <it>n</it>-cube <math><tmath>$(Q^k_n)$</tmath></math> using Lee distance. We feel that Lee distance is a natural metric for defining and studying a <math><tmath>$Q^k_n$</tmath></math>.</p><p>After defining a <math><tmath>$Q^k_n$</tmath></math> graph using Lee distance, we show how to find all disjoint paths between any two nodes. Given a sequence of radix <it>k</it> numbers, a function mapping the sequence to a Gray code sequence is presented, and this function is used to generate a Hamiltonian cycle.</p><p>Embedding the graph of a mesh and the graph of a binary hypercube into the graph of a <math><tmath>$Q^k_n$</tmath></math> is considered. Using a <it>k</it>-ary Gray code, we show the embedding of a <math><tmath>$k^{n_1}\times k^{n_2}\times \,\,\ldots \,\,\times k^{n_m}-$</tmath></math> dimensional mesh into a <math><tmath>$Q^k_n$</tmath></math> where <math><tmath>$n\,\,=\,\,\sum\nolimits_{i=1}^m {n_i}$</tmath></math>. Then using a single digit, 4-ary reflective Gray code, we demonstrate embedding a <it>Q</it><sub><it>n</it></sub> into a <math><tmath>$Q^4_{\lceil {n\over 2}\rceil}$</tmath></math>.</p><p>We look at how Lee distance may be applied to the problem of resource placement in a <math><tmath>$Q^k_n$</tmath></math> by using a Lee distance error-correcting code. Although the results in this paper are only preliminary, Lee distance error-correcting codes have not been applied previously to this problem.</p><p>Finally, we consider how Lee distance can be applied to message routing and single-node broadcasting in a <math><tmath>$Q^k_n$</tmath></math>. In this section we present two single-node broadcasting algorithms that are optimal when single-port and multi-port I/O is used.</p>

INDEX TERMS

K-ary n-cubes, Lee distance, error-correcting codes, Gray codes, Hamiltonian cycles, routing, broadcasting, embedding.

CITATION

B. Bose, Y. Kwon, Y. Ashir and B. Broeg, "Lee Distance and Topological Properties of k-ary n-cubes," in

*IEEE Transactions on Computers*, vol. 44, no. , pp. 1021-1030, 1995.

doi:10.1109/12.403718

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