Permutation routing in restricted access networks

In current massively parallel computers, permutation routing is often performed by a specialized router network. Each processing element (PE) has access to the router which is often implemented as a crossbar-based, multistage, circuit switched network. As the number of PEs grows, processors are grouped into clusters and share one port into the network. Because only one processor from every cluster can connect to the network at any given time, permutation routing requires at least as many cycles as there are processors in a cluster. It is shown that for a system with p clusters of q processors each, permutation routing can be achieved in exactly q network cycles. Four routing algorithms are then given