  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. We show that the constructions of certain types of optical queues,
including linear compressors, linear decompressors, and 2-to-1 FIFO
multiplexers, under a simple packet routing scheme and under the constraint of
a limited number of recirculations can be transformed into equivalent integer
representation problems under a corresponding constraint. Given $M$ and $k$,
the problem of finding an \emph{optimal} construction, in the sense of
maximizing the maximum delay (resp., buffer size), among our constructions of
linear compressors/decompressors (resp., 2-to-1 FIFO multiplexers) is
equivalent to the problem of finding an optimal sequence ${\dbf^*}_1^M$ in
$\Acal_M$ (resp., $\Bcal_M$) such that $B({\dbf^*}_1^M;k)=\max_{\dbf_1^M\in
\Acal_M}B(\dbf_1^M;k)$ (resp., $B({\dbf^*}_1^M;k)=\max_{\dbf_1^M\in
\Bcal_M}B(\dbf_1^M;k)$), where $\Acal_M$ (resp., $\Bcal_M$) is the set of all
sequences of fiber delays allowed in our constructions of linear
compressors/decompressors (resp., 2-to-1 FIFO multiplexers). In Part I, we
propose a class of \emph{greedy} constructions of linear
compressors/decompressors and 2-to-1 FIFO multiplexers by specifying a class
$\Gcal_{M,k}$ of sequences such that $\Gcal_{M,k}\subseteq \Bcal_M\subseteq
\Acal_M$ and each sequence in $\Gcal_{M,k}$ is obtained recursively in a greedy
manner. We then show that every optimal construction must be a greedy
construction. In Part II, we further show that there are at most two optimal
constructions and give a simple algorithm to obtain the optimal
construction(s).
