We study an interconnection network that we call 3Torus(m,n) obtained by pruning the 4m x 4n torus (of links) so that the resulting network is regular of degree 3. We show that 3Torus(m,n) retains many of the useful properties of tori (although, of course, there is a price to be paid due to the reduction in links). In particular: we show that 3Torus(m,n) is node-symmetric; we establish closed-form expressions on the the length of a shortest path joining any two nodes of the network; we calculate the diameter precisely; we obtain an upper bound on the average inter-node distance; we develop an optimal distributed routing algorithm; we prove that 3Torus(m,n) has connectivity 3 and is Hamiltonian; we obtain a precise expression for (an upper bound on) the wide-diameter; and we derive optimal one-to-all broadcast and personalized one-to-all broadcast algorithms under both a one-port and all-port communication model. We also undertake a preliminary performance evaluation of our routing algorithm. In summary, we find that 3Torus(m,n) compares very favourably with tori.
Stewart, I. (2014). Interconnection networks of degree three obtained by pruning two-dimensional tori. IEEE Transactions on Computers, 63(10), 2473-9340. https://doi.org/10.1109/tc.2013.139