Optical Routing of Uniform Instances in Tori.

We consider the problem of routing uniform communication instances in switched optical tori that use the wavelength division multiplexing (or WDM) approach. A communication instance is called uniform if it consists exactly of all pairs of nodes in the graph whose distance is equal to one from a specified set S = ( d 1 ; d 2 ,..., d k ). We give bounds on the optimal load induced on an edge for any uniform instance in a torus T n × n . When k = 1, we prove necessary and sufficient conditions on the value in S relative to n for the wavelength index to be equal to the load. When k ≥ 2, we show that for any set S , there exists an n 0 , such that for all n > n 0 , there is an optimal wavelength assignment for the communication instance specified by S on the torus T n × n . We also show an approximation for the wavelength index for any S and n . Finally, we give some results for rectangular tori.