Improved bounds for all optical routing

We consider the problem of routing in networks employing all optical routing technology. In these networks, messages travel in optical form and switching is performed directly on the optical signal. By using different wavelengths, several messages may use the same edge concurrently, However, messages assigned the same wavelength must use disjoint paths, or else be routed at separate rounds. No buffering at intermediate nodes is available. Thus, routing in this setting entails assigning wavelengths, paths, and time slots for the different messages. For arbitrary bounded degree networks, we show that any permutation can be routed efficiently in one round using at most O(log²n/β²) wavelengths, where β is the edge expansion of the network. This improves a quadratic factor on previous results, and almost matches the ω(l/β²) existential lower bound. We consider two of the more popular architectures for parallel computers. For bounded dimension arrays we give the first per-instance approximation algorithm. Given a limited number of wavelengths and a set of messages to be routed, the algorithm approximates to within polylogarithmic factors the optimal number of rounds necessary to route all messages. Previous results for arrays give only worst-case performance. Finally, we show that on the hypercube any permutation can be routed using only a constant number of wavelengths. The previous known bound was O(logre).