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Tight bounds for oblivious routing in the hypercube

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Abstract

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/d 3/2) 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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C. Kaklamanis was supported in part by NSF Grant NSF-CCR-87-04513. T. Tsantilas was supported in part by NSF Grants NSF-DCR-86-00379 and NSF-CCR-89-02500.

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Kaklamanis, C., Krizanc, D. & Tsantilas, T. Tight bounds for oblivious routing in the hypercube. Math. Systems Theory 24, 223–232 (1991). https://doi.org/10.1007/BF02090400

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  • DOI: https://doi.org/10.1007/BF02090400

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