Optical routing of uniform

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 = {dl, d2,..., d}. We give bounds on the optimal load induced on an edge for any uniform instance in a torus T,x,. 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 no, such that for all n no, there is an optimal wavelength assignment for the communication instance specified by S on the torus T,x,. We also show an approximation for the wavelength index for any S and n. Finally, we give some results for rectangular tori.