Expanders That Beat the Eigenvalue Bound: Explicit Construction and Applications

For every n and 0 > b > 1, we construct graphs on n nodes such that every two sets of size n^b share an edge, having essentially optimal maximum degree n^{1-b+o(1)}. We use them to explicitly construct a k round sorting algorithm using n^{1+1/k+o(1)} comparisons; a k round selection algorithm using n^{1+1/(2^k-1)+o(1)} comparisons; a depth 2 superconcentrator of size n^{1+o(1)}; and a depth k wide-sense nonblocking generalized connector of size n^{1+1/k+o(1)}. All of these results improve on previous constructions by factors of n^{Omega(1)}, and are optimal to within factors of n^{o(1)}. These results are based on an improvement to the extractor construction of Nisan & Zuckerman: our algorithm extracts asymptotically the optimal number of random bits from a defective random source using a small additional number of truly random bits.