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Lee Distance and Topological Properties of k-ary n-cubes
August 1995 (vol. 44 no. 8)
pp. 1021-1030

Abstract—In this paper, we consider various topological properties of a k-ary n-cube $(Q^k_n)$ using Lee distance. We feel that Lee distance is a natural metric for defining and studying a $Q^k_n$.

After defining a $Q^k_n$ graph using Lee distance, we show how to find all disjoint paths between any two nodes. Given a sequence of radix k numbers, a function mapping the sequence to a Gray code sequence is presented, and this function is used to generate a Hamiltonian cycle.

Embedding the graph of a mesh and the graph of a binary hypercube into the graph of a $Q^k_n$ is considered. Using a k-ary Gray code, we show the embedding of a $k^{n_1}\times k^{n_2}\times \,\,\ldots \,\,\times k^{n_m}-$ dimensional mesh into a $Q^k_n$ where $n\,\,=\,\,\sum\nolimits_{i=1}^m {n_i}$. Then using a single digit, 4-ary reflective Gray code, we demonstrate embedding a Qn into a $Q^4_{\lceil {n\over 2}\rceil}$.

We look at how Lee distance may be applied to the problem of resource placement in a $Q^k_n$ 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.

Finally, we consider how Lee distance can be applied to message routing and single-node broadcasting in a $Q^k_n$. In this section we present two single-node broadcasting algorithms that are optimal when single-port and multi-port I/O is used.

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Index Terms:
K-ary n-cubes, Lee distance, error-correcting codes, Gray codes, Hamiltonian cycles, routing, broadcasting, embedding.
Bob Broeg, Bella Bose, Younggeun Kwon, Yaagoub Ashir, "Lee Distance and Topological Properties of k-ary n-cubes," IEEE Transactions on Computers, vol. 44, no. 8, pp. 1021-1030, Aug. 1995, doi:10.1109/12.403718
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