On-line permutation routing on WDM all-optical networks

For a sequence (s₁, t₁), ..., (s_i, t_i), ... of routing requests with (s_i, t_i) arriving at time step i on the wavelength-division multiplexing (WDM) all-optical network, the on-line routing problem is to set-up a path s_i → t_i and assign a wavelength to the path in step i such that the paths set-up so far with the same wavelength are edge-disjoint. Two measures are important for on-line routing algorithms: the number of wavelengths used and the response time. The sequence (s₁,t₁), ..., (s_i, t_i), ... is called a permutation if each node in the network appears in the sequence at most once as a source and at most once as a destination. Let H_n be the n-dimensional WDM all-optical hypercube. We develop two on-line routing algorithms on H_n. Our first algorithm is a deterministic one which realizes any permutation by at most ┌3(n-1)/2┐ + 1 wavelengths with response time O(2ⁿ). The second algorithm is a randomized one which realizes any permutation by at most (3/2 + δ)(n-1) wavelengths, where δ can be any value satisfying δ ≥ 2/(n-1). The average response time of the algorithm is O(n(1 + δ)/δ). Both algorithms use at most O(n) wavelengths for the permutation on Hn. This improves the previous bound of O(n²).