Multi-hop all-to-all broadcast on WDM optical networks

It is known that the minimum number of wavelengths for realizing all-to-all broadcast (gossiping) in one-hop of optical routing on the ring (resp. the 2-dimensional torus) of N nodes is [[N²/4]/2] (resp. cN^1+1/2,c≈1/8). These numbers can be too large even for moderate values of N. One approach to reduce the number of wavelengths is to realize gossiping in multi-hops of routing. We prove that gossiping can be realized in k-hops by c_kN¹⁺¹k/(c_k≈1/2²⁺¹k/) wavelengths on the ring, c´N¹⁺¹(2k)/(c´≈1/4) wavelengths on the 2-dimensional torus, and c´^N1+1(3k)/ wavelengths on the 3-dimensional torus on the simple model: in the (j+1)st hop each node just forwards each message received in the jth hop to its next destinations. We also give the upper bounds on the numbers of wavelengths for gossiping in two-hops and three-hops for the ring, 2-dimensional torus, and 3-dimensional torus on the merge model: in the (j+1)st hop each node can merge different messages received in the jth hop into one and sends the merged message to its next destinations