Wide-Sense Nonblocking WDM Cross-Connects

We consider the problem of minimizing the number of wavelength interchangers in the design of wide-sense nonblocking crossconnects for wavelength division multiplexed (WDM) optical networks. The problem is modeled as a graph theoretic problem that we call dynamic edge coloring. In dynamic edge coloring the nodes of a graph are fixed but edges appear and disappear, and must be colored at the time of appearance without assigning the same color to adjacent edges. For wide-sense nonblocking WDM cross-connects with k input and k output fibers, it is straightforward to show that 2k−1 wavelength interchangers are always sufficient. We show that there is a constant c > 0 such that if there are at least ck wavelengths then 2k−1 wavelength interchangers are also necessary. This improves previous exponential bounds. When there are only 2 or 3 wavelengths available, we show that far fewer than 2k−1 wavelength interchangers are needed. However we also prove that for any ε > 0 and k > 1/2ε, if the number of wavelengths is at least 1/ε then 2(1−ε)k wavelength interchangers are needed.