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The Benes network has been used as a rearrangeable network for over 40 years, yet the uniform N(2 \log N-1) control complexity of the N \times N Benes is not optimal for many permutations. In this paper, we present a novel O(\log N) depth rearrangeable network, called KR-Benes, that is permutation-specific control-optimal. The KR-Benes routes every permutation with the minimal control complexity specific to that permutation and its worst-case complexity for arbitrary permutations is bounded by the Benes; thus, it replaces the Benes when considering control complexity/latency. We design the KR-Benes by first constructing a restricted 2 \log K +2 depth rearrangeable network called K{\hbox{-}}{\rm Benes} for routing K{\hbox{-}}{\rm bounded} permutations with control 2N \log K, 0 \leq K \leq N/4. We then show that the N \times N Benes network itself (with one additional stage) contains every K{\hbox{-}}{\rm Benes} network as a subgraph and use this property to construct the KR-Benes network. With regard to the control-optimality of the KR-Benes, we show that any optimal network for rearrangeably routing K{\hbox{-}}{\rm bounded} permutations must have depth 2 \log K + 2 and, therefore, the K{\hbox{-}}{\rm Benes} (and, hence, the KR-Benes) is optimal.
Benes network, rearrangeability, optimal control algorithm.

R. Kannan, "The KR-Benes Network: A Control-Optimal Rearrangeable Permutation Network," in IEEE Transactions on Computers, vol. 54, no. , pp. 534-544, 2005.
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