 
Summary: The Mimimal Number of Wavelengths Necessary for
Routing Permutations in All Optical Networks \Lambda
Yonatan Aumann y Yuval Rabani z
Abstract
We consider the problem of routing in networks employing all optical routing tech
nology. In these networks, messages travel in optical form and switching is performed
directly on the optical signal. By using different wavelengths, several messages may use
the same edge concurrently. However, messages traveling along the same wavelength
must use disjoint paths. No buffering at intermediate nodes is available.
We provide improved upper bounds for the manimal number of wavelengths neces
sary to route any permutation (or hrelation) in the optical setting. First we show that
on bounded degree networks any permutation can be routed efficiently using at most
O(log 2 n=fi 2 ) wavelengths, where fi is the edge expansion of the network. This improves
a quadratic factor on the previously known bound, and comes with a polylogarithmic
factor of the \Omega\Gammae =fi 2 ) existential lower bound. Then, we show that on the hypercube
any permutation can be routed using only a constant number of wavelengths.
Keywords: Interconnection networks, Routing, Optical routing, Combinatorial opti
mization.
\Lambda A preliminary version of this paper appeared in the proceedings of SODA 1995 as part of the paper
``improved bounds for all optical routing,'' by the same authors.
