On 1-rate and 2-rate multicast 3-stage Clos networks

Abstract

The 3-stage Clos network C(n, m, r) is considered as the most basic and popular multistage interconnection network which has been widely employed for data communications and parallel computing systems. Quite a lot of efforts has been put on the research of the 3-stage Clos network. Unfortunately, very little is known for the multirate multicast Clos network which is the most complicated case. Firstly a sufficient condition for 1-rate multicast networks to be SNB is given, from which a result for 2-rate multicast networks to be WSNB can easily be gotten. Furthermore, by using a reservation-scheme routing, more specific result for 2-rate multicast networks to be WSNB can be obtained for the case of one of them exceeding 1/2. §1 IntroductionThe well-known Clos network has been widely used for data communications and parallel computing systems to provide connections between two parts. Quite a lot of efforts has been put on the study of 3-stage Clos networks. The 3-stage Clos network C(n, m, r) is a 3-stage interconnection network and symmetric with respect to the center stage. C(n, m, r) consists of r (n × m)-crossbars (switches) in the first stage (or input stage), m (r × r)-crossbars in the second stage (or central stage), r (m × n)-crossbars in the third stage (or output stage). There exists exactly one link between every center crossbar and every input (output) crossbar.The Clos switching networks were firstly produced for the classical circuit switching, in which a request between an idle pair (input, output) should be connected by a path such that no link on the path is used by any other connection paths. A network is strictly nonblocking(SNB) if regardless of the routing of existing connections in the network, a new request is always routable. A network is wide-sense nonblocking(WSNB) if a new request is always routable as long as all connections are routed according to a given routing algorithm. The problem is to determine m o such that for all m ≥ m o , C(n, m, r) is SNB (or WSNB, respectively). Clos [5] proved that m o = 2n−1 for SNB. With m o for WSNB being still not completely settled, Chang, et al [3] have