KR-Benes网络：一种控制最优的可重排置换网络
The KR-Benes Network: A Control-Optimal Rearrangeable Permutation Network
摘要：
Benes网络已经被用作可替换网络超过40年，然而$N \times N$ Benes的统一$N(2 \log N-1)$控制复杂度对于许多排列不是最优的。在本文中，我们提出了一种新的$O(\log N)$的深度可重复网络称为KR-Benes，是{ \it置换特定的控制最优}。KR-Benes将具有最小控制复杂度{ \it specific}的{ \it every}置换路由到该置换，并且其对于任意置换的最坏情况复杂度由Benes限定;因此，当考虑控制复杂度/延迟时，它取代了Benes。我们设计KR-Benes通过首先构造一个被称为$K$ -Benes的受限$2 \log K +2$深度可重连网络，用于路由具有控制$2N \log K$的$K$ -有界置换，$0 \leq K \leq N/4$。然后，我们证明了$N \times N$ Benes网络本身（有一个额外的阶段）包含每个$K$ -Benes网络作为一个子图，并使用此属性来构建KR-Benes网络。关于KR-Benes的控制最优性，我们证明了任何最优的网络的路由$K$有界的排列必须有深度$2 \log K + 2$，因此$K$ -Benes（因此KR-Benes）是最优的。
英文摘要：
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 {\it permutation-specific control-optimal}. The KR-Benes routes {\it every} permutation with the minimal control complexity {\it 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 $K$-Benes for routing $K$-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 $K$-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 $K$-bounded permutations must have depth $2 \log K + 2$, and therefore the $K$-Benes (and hence the KR-Benes) is optimal.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
