Some permutation routing algorithms for low-dimensional hypercubes

AbstractOblivious permutation routing in binary d-cubes has been well studied in the literature. In a permutation routing, each node initially contains a packet with a destination such that all the 2d destinations are distinct. Kaklamanis et al. (Math. Syst. Theory 24 (1991) 223–232) used the decomposability of hypercubes into Hamiltonian circuits to give an asymptotically optimal routing algorithm. The notion of “destination graph” was first introduced by Borodin and Hopcroft to derive lower bounds on routing algorithms. This idea was recently used by Grammatikakis et al. (Proceedings of the Advancement in Parallel Computing, Elsevier, Amsterdam, 1993) to construct many–one routing algorithms for the binary 2-cube and 3-cube. In the present paper, further theoretical development is made along this line. It is then applied to obtain algorithms for binary d-cubes with d up to 12, which compare favorably with the above-mentioned “Hamiltonian circuit” algorithm. Some results on t-nary cubes with t⩾3 are also obtained