Lee distance and topological properties of k-ary n-cubes

In this paper, we consider various topological properties of a k-ary n-cube (Q_n^k) using Lee distance. We feel that Lee distance is a natural metric for defining and studying a Q_n^k. After defining a Q_n^k 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_n^k is considered. Using a k-ary Gray code, we show the embedding of a k(n₁)×k(n₂)×...×k(n_m )-dimensional mesh into a Q_n^k where n=Σ_i=l^mn_i. Then using a single digit, 4-ary reflective Gray code, we demonstrate embedding a Q_n into Q_[n/2]⁴. We look at how Lee distance may be applied to the problem of resource placement in a Q_n ^k 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_n^k. In this section we present two single-node broadcasting algorithms that are optimal when single-port and multi-port I/O is used