Constructions of Optical Queues With a Limited Number of Recirculations--Part II: Optimal Constructions

Abstract

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.