The KR-Benes Network: A (almost) Control-Optimal Rearrangeable Permutation Network

In this paper, we solve a longstanding open problem on the design of optimal rearrangeable networks. Networks such as the Benes [1] have been used for over 40 years, yet the uniform N(2 logN − 1) control complexity of the N × N Benes is not optimal for many permutations. To the best of our knowledge, there are no results in the literature that improve this control complexity. In this paper, we present a novel N × N rearrangeable network called KR-Benes that is permutation-specific control-optimal. The KR-Benes routes every permutation with the (conjectured) minimal control complexity specific to that permutation and is provably within an additive linear factor (N ) of the optimal for that permutation (corresponding to controlling one extra stage of switching elements). In fact, the worst-case complexity of the KR-Benes for arbitrary permutations is bounded by the Benes; thus it replaces the Benes when considering control complexity/latency. We construct the KR-Benes by first designing a restricted rearrangeable network called KBenes for optimally routing K-bounded permutations with control 2N logK , 0 ≤ K ≤ N/2, using a simple modification of the Benes network looping algorithm. In our main result, we show that the N × N Benes network itself (with 2 additional stages) contains every K-Benes network as a subgraph. This then becomes our control-optimal KR-Benes network. With regard to the optimality of the KR-Benes, we first show that any optimal network for rearrangeably routing K-bounded permutations must have depth at least 2 logK + 1, 0 ≤ K ≤ N/4 (depth at least 2 logK for K = N/2). Each K-Benes network is shown to have a depth of one more stage than this provable lower bound. We then provide a strong conjecture that the optimal network, in fact has 2 logK+2 stages, and therefore the K-Benes (and hence the KR-Benes) is control-optimal.