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

TL;DR

This paper investigates optimal optical queue constructions using fiber delay lines with limited recirculations, providing a simple algorithm based on Euclid's algorithm to identify at most two optimal solutions depending on the gcd of key parameters.

Contribution

It introduces a method to determine at most two optimal constructions for optical queues with limited recirculations, utilizing pairwise comparison and Euclid's algorithm.

Findings

At most two optimal constructions exist for given parameters.

The optimal construction depends on the gcd of M and k.

A simple Euclid's algorithm-based method identifies optimal solutions.

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.