Uniform Multi - hop All - to - All Optical

We consider all-to-all routing problem in an optical ring network that uses the wavelength-division multiplexing (WDM). Since one-hop, all-to-all optical routing in a WDM optical ring of n nodes needs ! w (Cn; IA; 1) = d 1 2 b n 2 4 ce wavelengths which can be too large even for moderate values of n, we consider in this paper j-hop implementations of all-to-all routing in a WDM optical ring, j 2. From among the possible routings we focus our attention on uniform routings, in which each node of the ring uses the same communication pattern. We show that there exists a uniform 2-hop, 3-hop, and 4-hop implementation of all-to-all routing that needs at most n+3 3 p n+16 5 + n 4 , n 2 3 q dn=4e+4 5 , and n+16 2 4 p d n 4 e + 8 wavelengths, respectively. These values are within mul-tiplicative constants of lower bounds.