Uniform Multi-hop All-to-All Optical Routings in Rings

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 \(\stackrel{\rightarrow}{w}(C_n,I_A,1)= \lceil \frac{1}{2}\lfloor \frac{n^2}{4}\rfloor\rceil\) 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 \(\frac{n+3}{3}\sqrt{\frac{n+16}{5}} + \frac{n}{4}\), \( \frac{n}{2}\sqrt[3]{\frac{\lceil n/4 \rceil+4}{5}}\), and \(\frac{n+16}{2}\sqrt[4]{\lceil \frac{n}{4}\rceil+8}\) wavelengths, respectively. These values are within multiplicative constants of lower bounds.