Optimal placement of wavelength converters in trees and trees of rings

In wavelength-routed optical networks, wavelength converters can potentially reduce the requirement on the number of wavelengths. The problem of placing a minimum number of wavelength converters in a WDM network so that any routing can be satisfied using no more wavelengths than if there were wavelength converters at every node was raised in Wilfong et al. (1998) and shown to be NP-complete in general WDM networks. Recently, it was proved in Kleinberg et al. (1999) that this problem is as hard as the well-known minimum vertex cover problem. In this paper, we further their study in two topologies that are of more practical concrete relevance to the telecommunications industry: trees and tree of rings. We show that the optimal wavelength converter placement problem in these two practical topologies are tractable. Efficient polynomial-time algorithms are presented.