Analyzing nonblocking switching networks using linear programming (duality)

The main task in analyzing a switching network design (including circuit-, multirate-, and photonic-switching) is to determine the minimum number of some switching components so that the design is non-blocking in some sense (e.g., strict- or wide-sense). We show that, in many cases, this task can be accomplished with a simple two-step strategy: (1) formulate a linear program whose optimum value is a bound for the minimum number we are seeking, and (2) specify a solution to the dual program, whose objective value by weak duality immediately yields a sufficient condition for the design to be non-blocking. We illustrate this technique through a variety of examples, ranging from circuit to multirate to photonic switching, from unicast to f-cast and multicast, and from strict- to wide-sense non-blocking. The switching architectures in the examples are of Clos-type and Banyan-type, which are the two most popular architectural choices for designing non-blocking switching networks. To prove the result in the multirate Clos network case, we formulate a new problem called DYNAMIC WEIGHTED EDGE COLORING which generalizes the DYNAMIC BIN PACKING problem. We then design an algorithm with competitive ratio 5.6355 for the problem. The algorithm is analyzed using the linear programming technique. We also show that no algorithm can have competitive ratio better than 4 - O(log n/n) for this problem. New lower- and upper-bounds for multirate wide-sense non-blocking Clos networks follow, improving upon a couple of 10-year-old bounds on the same problem.