Semi-Explicit Construction of Linear Size Concentrators and Superconcentrators

Concentration and superconcentration operations play a central role in the constructions of nonblocking switching networks. This was established in a seminal effort by Bassalygo and Pinsker who showed that an n-input nonblocking switching network can be constructed with O(n log n) crosspoints if concentrators can be constructed with O(n) crosspoints. Starting with the pioneering work of Margulis, concentrators and superconcentrators with O(n) crosspoints were reported in the literature, but the constants in the crosspoint complexities of these concentrators are considerably large. In this paper, we extend the probabilistic constructions of Pinsker, Bassalygo and Pippenger to obtain semi-explicit constructions of concentrators and superconcentrators with fewer crosspoints. More specifically, we give a semi-explicit version of Pinsker’s bounded capacity concentrator to show that there exists a concentrator with at most 25n crosspoints. This improves Pinsker’s original bound of 29n crosspoints. Next we optimize Bassalygo’s concentrator construction to prove that there exists a concentrator with 19n crosspoints, improving his bound of 20n crosspoints. We then give a semi-explicit (n, 3n/4, n/2)-concentrator cosntruction with 4.5n crosspoints, and use this bounded capacity concentrator to obtain an n-superconcentrator with 40n crosspoints. Finally, we show that Bassalygo’s superconcentrator construction can be optimized to obtain a superconcentrator with 33n crosspoints.