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

Abstract

We consider the problem of minimizing the number of ADMs in optical networks. All previous theoretical studies of this problem dealt with the off-line case, where all the lightpaths are given in advance. In a real-life situation, the requests (lightpaths) arrive at the network on-line, and we have to assign them wavelengths so as to minimize the switching cost. This study is thus of great importance in the theory of optical networks. We present an on-line algorithm for the problem, and show its competitive ratio to be $\frac{7}{4}$. We show that this result is best possible in general. Moreover, we show that even for the ring topology network there is no on-line algorithm with competitive ratio better than $\frac{7}{4}$. We show that on path topology the competitive ratio of the algorithm is $\frac{3}{2}$. This is optimal for this topology. The lower bound on ring topology does not hold when the ring is of bounded size. We analyze the triangle topology and show a tight bound of $\frac{5}{3}$ for it. The analyzes of the upper bounds, as well as those for the lower bounds, are all using a variety of proof techniques, which are of interest by their own, and which might prove helpful in future research on the topic.