Some permutation routing algorithms for low-dimensional hypercubes

Abstract

Oblivious 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 2(d) 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 greater than or equal to 3 are also obtained. (C) 2002 Published by Elsevier Science B.V.