The necessary conditions for Clos-type nonblocking multicast networks

Efficient interconnection 'networks are critical in providing low latency, high bandwidth communication in parallel and distributed computing systems with hundreds or thousands of processors. The well-known Clos network or v(m, n, r) network can be extended to provide full one-to-many or multicast capability. In this paper, we consider several typical routing control strategies for Clos-type nonblocking multicast networks and derive the necessary conditions under which this type of network is nonblocking for arbitrary multicast assignments in the strict sense as well as under these control strategies. The necessary conditions obtained are represented as the number of middle stage switches m greater than or equal to Theta(n logr/loglogr). These results match the sufficient nonblocking condition for the currently best available explicitly constructed, constant stage nonblocking multicast network [8], [9], and provide a basis for the optimal design of this type of multicast network.