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<p><b>Abstract</b>—We consider the problems of routing and sorting on a de Bruijn network. First, we show that any deterministic oblivious routing scheme for permutation routing on a <it>d</it>-ary de Bruijn network with N = d<super><it>n</it></super> nodes, in the worst case, will take <tmath>$\Omega (\sqrt N)$</tmath> steps under the <it>single-port</it> model. This improves the existing lower bounds provided <it>d</it> is not a constant. We also show that the lower bound is indeed a tight one. Second, we present a deterministic nonoblivious permutation routing algorithm which runs in <it>O</it>(<it>d</it>·<it>n</it><super>2</super>) time on a <it>d</it>-ary de Bruijn network with N = d<super><it>n</it></super> nodes. This algorithm is currently the fastest known nonoblivious deterministic routing algorithm for de Bruijn networks of arbitrary degree. Finally, we present an efficient general sorting algorithm for the de Bruijn networks of arbitrary degree. This algorithm is the best sorting algorithm known so far. It runs in <it>O</it>((log <it>d</it>) ·<it>d</it>·<it>n</it><super>2</super>) time for directed de Bruijn network with d<super><it>n</it></super> nodes, degree <it>d</it>, and diameter <it>n</it>. As a corollary, we show that on a binary de Bruijn network of <it>N</it> nodes, our sorting scheme requires at most 2 log<super>2</super><it>N</it> steps.</p>
Oblivious routing, nonoblivious routing, permutation routing, parallel sorting, de Bruijn network.

D. S. Wei and D. F. Hsu, "Efficient Routing and Sorting Schemes for de Bruijn Networks," in IEEE Transactions on Parallel & Distributed Systems, vol. 8, no. , pp. 1157-1170, 1997.
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