On wavelength assignment in WDM optical networks

We address the problem of assigning wavelengths to paths (connections) in optical wavelength division multiplexed networks. The problem is formulated as follows: given the physical topology of a network with each edge of two opposite directed fiber links and a set of directed paths with no more than L paths over any fiber link, we assign a wavelength to each path in such a way that no two paths are assigned the same wavelength if they share a directed physical link. In this paper, we first prove that the problem is NP-complete for arbitrary network topologies. Our NP-completeness result is obtained through a polynomial time reduction from the graph k-colorability problem. This reduction implies that no polynomial time algorithm can solve the problem with the number of wavelengths bounded by a constant times L for the class of network topologies including meshes. We then consider tree topologies. For star networks (i.e. the length of any path is bounded by two), we give a polynomial time algorithm that requires L wavelengths. For trees with path lengths larger than two, we show that the problem is NP-complete and present a heuristic algorithm based on the polynomial time algorithm for star topologies. Our heuristic algorithm may require 2L wavelengths in the worst-case, but the simulation result shows that the average-case performance significantly outperforms the worst-case bound. This suggests that fewer excess wavelengths are required when most of the load is local.