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 multi-plexing (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 speciied set S = fd1; d2; : : : ; d k g. We give bounds on the optimal load induced on an edge for any uniform instance in a torus Tnn. When k = 1, we prove necessary and suucient 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 n0, such that for all n > n0, there is an optimal wavelength assignment for the communication instance speciied by S on the torus Tnn. We also show an approximation for the wavelength index for any S and n. Finally, we give some results for rectangular tori.