Data Rearrangement between Radix-k and Lee Distance Gray Codes in k-ary n-cubes

The k-ary n-cube is one of the popular topologies for interconnecting processors in multicomputers. This paper studies the difference in communication requirements between two Lee distance Gray codes when moving data from processors in normal radix k order to those in Gray code order in k-ary n-cube networks. Algorithms for k-ary to Gray code conversion, and vice versa, in k-ary n-cube networks are described under various channel constraints, i.e., one-port and all-port communication assumptions. The minimum length path routing algorithm for nonreflective Gray code requires roughly M(k/4) and (n − 1) M(k/4) steps for data element transfers under all-port communication and one-port communication, respectively, for M elements per node. It is also shown that using a nonminimum length path routing algorithm, the number of steps for data element transfers can be halved. Lower bounds for the number of element transfers are derived, and the proposed algorithm using nonminimum length paths under one-port communication is shown to be near optimal.