  BCube is a highly scalable and cost-effective networking topology, which has
been widely applied to modular datacenters. Optical technologies based on
Wavelength Division Multiplexing (WDM) are gaining popularity for Data Center
Networks (DCNs) due to their technological strengths such as low communication
latency, low power consumption, and high link bandwidth. Therefore, it is worth
investigating optical techniques into the BCube architecture for future DCNs.
For this purpose, we study the forwarding and optical indices in an all-optical
BCube network. Consider an all-optical BCube network in which every host sets
up a connection with every other host. The optical index is the minimum number
of wavelengths required by the network to support such a host-to-host traffic,
under the restriction that each connection is assigned a wavelength that
remains constant in the network. A routing is a set of directed paths specified
for all host pairs. By defining the maximum link load of a routing as the
maximum number of paths passing through any link, the forwarding index is
measured to be the minimum of maximum link load over all possible routings. The
forwarding index turns out to be a natural lower bound of the optical index. In
this paper, we first compute the forwarding index of an all-optical BCube
network. Then, we derive an upper bound of the optical index by providing an
oblivious routing and wavelength assignment (RWA) schemes, which attains the
lower bound given by the forwarding index in some small cases. Finally, a
tighter upper bound is obtained by means of the chromatic numbers in Graph
Theory.
