Crosspoint complexity of sparse crossbar concentrators

A sparse crossbar (n,m, c)-concentrator is a bipartite graph with n inputs and m outputs in which any c or fewer inputs can be matched with an equal number of outputs, where c is called its capacity. Sparse crossbar concentrators play a key role in multiplexing signals in communication systems and constructing permutation and generalized connection networks. In this paper we present a number of new results on the crosspoint complexity of such concentrators. First, we describe an (n,m,m)concentrator construction, called a fat-and-slim crossbar whose crosspoint complexity matches Nakamura-Masson’s lower bound for any given n and m, thereby removing a restriction on the number of inputs and number of outputs imposed by Masson’s binomial concentrator construction. Second, we present a (2m,m)-concentrator with crosspoint complexity matching Nakamura-Masson’s lower bound and with fanout which is nearly the half of the fanout of the binomial concentrator. Third, we derive a new lower bound on the crosspoint complexity of sparse crossbar (n,m, c)-concentrators. This bound closely follows Nakamura-Masson’s lower bound, but unlike that bound, it is a simple rational function of n,m and c, and hence, can be computed very easily. We show that this bound is attainable when n − m ≤ c ≤ √m. Finally, we give an extension of this bound to cascades of sparse crossbar concentrators.