Wavelength Routing of Uniform Instances in Optical Rings

We consider the problem of routing uniform communication instances in switched optical rings 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 = fd1;d2; : : : ;dkg. When k = 1 or 2, we prove necessary and sufficient conditions on the values in S relative to n for the optimal wavelength index to be equal to the optimal load. When k = 2, we show that for any uniform instance specified by fd1;d2g, there is an optimal wavelength assignment on the ring Rn, if n > (d1 2)d1 +(d1 1)d2. For general k, we show a (3=2)-approximation for the optimal wavelength index; this is the best possible for arbitrary S. We also give an optimal wavelength assignment if n is large enough. Finally, we show some results for uniform permutations.