Probabilistic routing in wavelength-routed multistage, hypercube, and Debruijn networks

Optical networks based on wavelength division multiplexing (WDM) and wavelength routing are considered to be potential candidates for the next generation of wide area networks. One of the main issues in these networks is the development of efficient routing algorithms which require a minimum number of wavelengths. We focus on the permutation routing problem in multistage WDM networks which we call 2-multinets. We present a simple, oblivious probabilistic approach which solves the permutation routing problem on 2-multinets with very high probability (in the usual theoretical sense) using O(log²N/loglogN) wavelengths, where N is the number of nodes in the network, thereby improving the previous result due to Pankaj and Gallager (1995) that requires O(log³ N) wavelengths. Our approach is advantageous and practical as it is simple, oblivious, and suitable for centralized as well as distributed implementations. We also note that O(logN) wavelengths will suffice with good probabilistic guarantee for the case of dynamic permutation routing where requests arrive and terminate without any relation to each other. The above results are for networks with wavelength converters and we show that the use of converters can be eliminated at the expense of a factor of log N more wavelengths. We also show how our approach can be used to solve the dynamic permutation routing problem well (in practice), using O(1) wavelengths on the hypercube and O(logN) wavelengths on the Debruijn network. These improve the previous known bounds of O(logN) and O(log²N), respectively