具有有限循环次数的光学器件的结构--II部分：最佳结构
Constructions of Optical Queues With a Limited Number of Recirculations--Part II: Optimal Constructions
摘要：
全光分组交换网络中的一个主要问题是光缓存的缺乏，而光缓存的一种可行的构造技术是采用光交叉开关和光纤延迟线（SDL）。在这篇由两部分组成的论文中，我们考虑了通过光开关和光纤延迟线的有限数量的再循环的光队列的SDL结构。这一问题源于实际可行性考虑。在第一部分中，我们提出了一类贪婪的某些类型的光队列的建设，包括线性压缩器，线性解压缩器，和2比1的FIFO复用器，并已表明，我们以前的这些类型的光队列的建设中的每一个最佳的结构在有限数量的再循环的约束下必须是一个贪婪的建设。在本文的第二部分中，我们进一步证明了最多有两个最优结构，并给出了一个简单的算法来获得最优结构。第II部分的主要思想是使用\emph{pairwise comparison}来删除序列$\dbf_1^M\in \Gcal_{M,k}$，使得对于某些${\dbf'}_1^M\in \Gcal_{M,k}$，可以使用$B(\dbf_1^M;k)<B({\dbf'}_1^M;k)$。令我们惊讶的是，用于获得最佳构造的简单算法与用于找到两个整数的最大公约数（gcd）的著名的\emph{Euclid's algorithm}有关。特别是，我们表明，如果$\gcd(M,k)=1$，那么只有一个最佳的建设，如果$\gcd(M,k)=2$，那么有两个最佳的建设，如果$\gcd(M,k)\geq 3$，那么最多有两个最佳的建设。
英文摘要：
One of the main problems in all-optical packet-switched networks is the lack of optical buffers, and one feasible technology for the constructions of optical buffers is to use optical crossbar Switches and fiber Delay Lines (SDL). In this two-part paper, we consider SDL constructions of optical queues with a limited number of recirculations through the optical switches and the fiber delay lines. Such a problem arises from practical feasibility considerations. In Part I, we have proposed a class of greedy constructions for certain types of optical queues, including linear compressors, linear decompressors, and 2-to-1 FIFO multiplexers, and have shown that every optimal construction among our previous constructions of these types of optical queues under the constraint of a limited number of recirculations must be a greedy construction. In Part II, the present paper, we further show that there are at most two optimal constructions and give a simple algorithm to obtain the optimal construction(s). The main idea in Part II is to use \emph{pairwise comparison} to remove a sequence $\dbf_1^M\in \Gcal_{M,k}$ such that $B(\dbf_1^M;k)<B({\dbf'}_1^M;k)$ for some ${\dbf'}_1^M\in \Gcal_{M,k}$. To our surprise, the simple algorithm for obtaining the optimal construction(s) is related to the well-known \emph{Euclid's algorithm} for finding the greatest common divisor (gcd) of two integers. In particular, we show that if $\gcd(M,k)=1$, then there is only one optimal construction; if $\gcd(M,k)=2$, then there are two optimal constructions; and if $\gcd(M,k)\geq 3$, then there are at most two optimal constructions.
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