Expanders that Beat the Eigenvalue Bound: Explicit Construction

For every n and 0 < δ < 1, we construct graphs on n nodes such that every two sets of size n share an edge, having essentially optimal maximum degree n1−δ+o(1). Using known and new reductions from these graphs, we explicitly construct: 1. A k round sorting algorithm using n comparisons. 2. A k round selection algorithm using n −1)+o(1) comparisons. 3. A depth 2 superconcentrator of size n. 4. A depth k wide-sense nonblocking generalized connector of size n. All of these results improve on previous constructions by factors of n, and are optimal to within factors of n. These results are based on an improvement to the extractor construction of Nisan & Zuckerman: our algorithm extracts an asymptotically optimal number of random bits from a defective random source using a small additional number of truly random bits.