Wavelength Assignment Problem on All-Optical Networks with k Fibres per Link

Abstract

Given a (possibly directed) network, the wavelength assignment problem is to minimize the number of wavelengths that must be assigned to communication paths so that paths sharing an edge are assigned different wavelengths. Our generalization to multigraphs with k parallel edges for each link (k fibres per link, with switches at nodes) may be of practical interest. While the wavelength assignment problem is NP-hard, even for a single fibre, and even in the case of simple network topologies such as rings and trees, the new model suggests many nice combinatorial problems, some of which we solve. For example, we show that for many network topologies, such as rings, stars, and specific trees, the number of wavelengths needed in the k-fibre model is less than 1/k fraction of the number required for a single fibre. We also study the existence and behavior of a gap between the minimum number of wavelengths and the natural lower bound of network congestion, the maximum number of communication paths sharing an edge. For optical stars (any size) while there is a 3/2 gap in the single fibre model, we show that with 2 fibres the gap is 0, and present a polynomial time algorithm that finds an optimal assignment. In contrast, we show that there is no fixed constant k such that for every ring and every set of communication paths the gap can be eliminated. A similar statement holds for trees. However, for rings, the gap can be made arbitrarily small, given enough fibres. The gap can even be eliminated, if the length of communication paths is bounded by a constant. We show the existence of anomalies: increasing the number of fibres may increase the gap.