Tight bounds for oblivious routing in the hypercube

We prove that in anyN-node communication network with maximum degreed, any deterministic oblivious algorithm for routing an arbitrary permutation requires Ω(√N/d) parallel communication steps in the worst case. This is an improvement upon the Ω(√N/d3/2) bound obtained by Borodin and Hopcroft. For theN-node hypercube, in particular, we show a matching upper bound by exhibiting a deterministic oblivious algorithm that routes any permutation in Θ(√N/logN) steps. The best previously known upper bound was Θ(√N). Our algorithm may be practical for smallN (up to about 214 nodes).

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