Limited wavelength conversion in all-optical tree networks

Let T be a symmetric directed tree, i.e., a tree with each edge viewed as two opposite directed links. We consider the problem of routing arbitrary sets of connection requests in T. In all-optical communication tree networks with WDM (wavelength-division multiplexing) this is equivalent to color a given set of directed paths so that no two directed paths of the same color use the same link of T. Let W be the number of available wavelengths. The load, that is, the maximum number of directed paths passing through a link of T cannot exceed W. If there is no wavelength conversion available then each request (directed path) is restricted to a single wavelength and it is known that the minimum number of colors needed to color any set of directed paths in a tree is lower bounded away from the load L of the paths on the tree; moreover, no algorithm is known that uses a number of colors less than 5L/3. The presence of wavelength converters allows to overcome this limits. The complexity of converters is measured by the maximum degree, which represents the maximum number of possible conversions of any given wavelength. We show that it is possible to route any set of requests of load L less than or equal to W(1 - epsilon), where epsilon depends only on the degree of the converters. Moreover. we show that in any tree it is possible to route any set of requests of load L less than or equal to W with converters of degree O(root W). In case of trees containing at most one node of degree greater than or equal to 4 we show that converters of degree 3 already allow to route any set of requests of load L less than or equal to W.