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 dissertation, 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 our previous work, we have shown that the \emph{effective} maximum delay of a linear compressor/decompressor and the effective buffer size of a 2-to-1 FIFO multiplexer in our constructions are equal to
the \emph{maximum representable integer} $B(\dbf_1^M;k)$ with respect to $\dbf_1^M$ and $k$ (see (1.4) in Chapter 1 for the definition of $B(\dbf_1^M;k)$), where $\dbf_1^M=(d_1,d_2,\ldots,d_M)$ is the sequence of the delays of the $M$ fibers used in our constructions and $k$ is the maximum number of times that a packet can be routed through the $M$ fibers. Furthermore, we have proposed a class of greedy constructions $\Gcal_{M,k}$ for 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 other words, we have shown that to find an optimal construction, it suffices to find an optimal sequence ${\dbf^*}_1^M\in \Gcal_{M,k}$ such that
$B({\dbf^*}_1^M;k)=\max_{\dbf_1^M\in \Gcal_{M,k}}B(\dbf_1^M;k)$.

In this dissertation, 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 this dissertation 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.

在全光封包交換網路 (all-optical packet-switched networks) 中，建造光封包的緩衝器是一個重要的課題，而目前已知可行的方法是利用光交換機 (optical switches) 與光纖延遲線 (fiber delay lines) 來建造光封包的緩衝器。在這篇論文裡，我們考慮的問題是如何建造有繞行次數限制之光佇列 (optical queues)。這樣的一個繞行次數限制是來自實作上的考量。由先前的研究成果，我們已經知道在有繞行次數限制的情況下，光學線性壓縮器 (optical linear compressors) 與光學線性解壓縮器 (optical linear decompressors) 的最大有效延遲時間 (effective maximum delay)，以及雙輸入埠與單輸出埠之先進先出光學多工器 (optical 2-to-1 FIFO multiplexers) 的有效緩衝區容量 (effective buffer size)，都等於一個最大可表示整數$B(\dbf_1^M;k)$，其中$\dbf_1^M=(d_1,d_2,\ldots,d_M)$為光纖延遲線長度的數列，$k$為光封包繞行此$M$條光纖的次數限制 ($B(\dbf_1^M;k)$的數學定義請參閱論文第一章的(1.4)式)。另外，我們已經知道光學線性壓縮器、光學線性解壓縮器、以及雙輸入埠與單輸出埠之先進先出光學多工器的最佳建造方式，必定可由一個貪婪建造法 (greedy constructions) 得到，換句話說，要找到上述光佇列的最佳建造方式 ，我們只需在貪婪建造法所構成的集合$\Gcal_{M,k}$裡找出一個${\dbf^*}_1^M$使得$B({\dbf^*}_1^M;k)=\max_{\dbf_1^M\in\Gcal_{M,k}}B(\dbf_1^M;k)$即可。

在這篇論文裡，我們證明上述光佇列最多存在兩種最佳建造方式，並且我們提出一個簡單的演算法以得到這些最佳建造方式。我們主要的做法是透過配對比較 (pairwise comparison) 來移除$\Gcal_{M,k}$中比較不好的建造方式。出人意料地，我們的演算法與輾轉相除法 (Euclid's algorithm) 有關。我們證明了如果$\gcd(M,k)=1$，那麼只存在一個最佳建造方式；如果$\gcd(M,k)=2$，那麼我們有兩種最佳建造方式；如果$\gcd(M,k)=3$，那麼最多只存在兩種最佳建造方式。

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