Tight bounds for depth-two superconcentrators

We show that the minimum size of a depth-two N-superconcentrator is Θ(Nlog²N/loglogN). Before this work, optimal bounds were known for all depths except two. For the upper bound, we build superconcentrators by putting together a small number of disperser graphs; these disperser graphs are obtained using a probabilistic argument. We present two different methods for showing lower bounds. First, we show that superconcentrators contain several disjoint disperser graphs. When combined with the lower bound for disperser graphs due to Kovari, Sos and Turan, this gives an almost optimal lower bound of Ω(N(log N/loglog N)²) on the size of N-superconcentrators. The second method, based on the work of Hansel (1964), gives the optimal lower bound. The method of the Kovari, Sos and Turan can be extended to give tight lower bounds for extractors, both in terms of the number of truly random bits needed to extract one additional bit and in terms of the unavoidable entropy loss in the system. If the input is an n-bit source with min-entropy κ and the output is required to be within a distance of E from uniform distribution, then to extract even a constant number of additional bits, one must invest at least log(n-κ)+2 log(1/ε)-O(1) truly random bits; to obtain m output bits one must invest at least m-κ+2 log(1/ε)-O(1). Thus, there is a loss of 2 log(1/ε) bits during the extraction. Interestingly in the case of dispersers this loss in entropy is only about loglog(1/ε)