Optimal On-Line Colorings for Minimizing the Number of ADMs in Optical Networks

Optical wavelength-division multiplexing (WDM) is today the most promising technology, that enables us to deal with the enormous growth of traffic in communication networks. A communication between a pair of nodes is using lightpaths. In graph-theoretic terms, a lightpath is a simple path in the network, with a color assigned to it. Given a WDM network comprising of optical nodes and a set of lightpaths, the wavelength assignment task is to assign a wavelength to each lightpath, such that lightpaths that share an edge will get different colors. The switching cost is defined as follows. Each lightpath uses two ADM's, one at each endpoint. If two adjacent lightpaths are assigned the same wavelength, then they can use the same ADM. An ADM may be shared by at most two lightpaths. The objective is to find an assignment of colors to the lightpaths that minimizes the total number of ADMs. The problem was introduced in [1] for ring topology. Approximation algorithm for ring topology with approximation ratio of 3/2 was presented in [2], and was improved in [3, 4] to 10/7 + epsiv and 10/7 , respectively. All these - as well as other - studies concerning this problem dealt with the off-line case, where all the lightpaths are given in advance. Naturally, a more realistic problem to study is the on-line version of the problem, where the requests (lightpaths) arrive at the network on-line, and we have to assign them wavelengths so as to minimize the switching cost. We present an on-line algorithm whose performance is at most 7/4 of the optimal solution (i.e., its competitive ratio - see [5] - is 7/4) for any network topology. We prove that no algorithm has a competitive ratio better than 7/4 even if the topology is a ring. We show that the same algorithm has a competitive ratio of 3/2 in path topologies, and that also this bound is optimal. The lower bound on ring topology does not hold when the ring is of a bounded size; we study the triangle topology, and show a tight bound of 5/3 for the competitive ratio on this topology, using another algorithm. In the analyses of the upper bounds and the lower bounds we use a variety of proof techniques, which are of interest by their own, and which might prove helpful in future research on the topic.