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. We present a number of new results on the crosspoint complexity of such concentrators. First, we describe a sparse crossbar (n, m, m)-concentrator whose crosspoint complexity matches Nakamura-Masson´s (1982, 1977) lower bound for any given n and m. Second, we present a sparse crossbar (2m, m, m)-concentrator with crosspoint complexity also matching Nakamura-Masson´s lower bound, and with fixed fan-in and nearly fixed fan-out. Third, we derive an easily computable lower bound on the crosspoint complexity of sparse crossbar (n, m, c)-concentrators. Finally, we show that this bound is attainable within a factor of two when n-m⩽c⩽[m/c]