When analyzing a nonblocking switching network, the typical problem is to find a route for a new request through the network without disturbing existing routes. By solving this problem, we can derive how many hardware components of a certain type (Banyan planes in a multi-log network, for instance) are needed for the network to be nonblocking. This scenario appears in virtually all combinations of switching environments: strictly, widesense or rearrangeably nonblocking, unicast or multicast switching, and circuit, multirate, or photonic switching. In this paper, we show that the König–Egevarý theorem is a very good tool which helps solve the above prototypical problem. The idea is to somehow "represent" the potential blocking connections as edges of a bipartite graph. The maximum number of blocking connections roughly corresponds to the size of a maximum matching in that bipartite graph. The size of any vertex cover, by the König–Egevarý theorem, is an upper bound on the maximum number of blocking connections. Thus, by specifying a small vertex cover, we can derive the sufficient number of hardware components for the network to be nonblocking. We illustrate the technique by analyzing crosstalk-free and non-crosstalk-free widesense nonblocking multicast multi-log networks. Particularly, for the first time in the literature we derive conditions for the d-ary multi-log network to be crosstalk-free multicast widesense nonblocking under the window algorithm for any given window size. Several by-products follow from our approach and results. Firstly, our results allow for computing the best window size minimizing the fabric cost, showing that the multi-log network is a good candidate for crosstalk-free multicast switching architectures. Secondly, the analytical approach also gives a much simpler proof of the known case when the network is not required to be crosstalk-free. Thirdly, we show that several known results for the multi-log multicast networks under the so-called fanout constraints are simple corollaries of our results.
Dynamic Approximate Vertex Cover and Maximum Matching