Expanders that Beat the Eigenvalue Bound : Explicit

For every n and 0 < < 1, we construct graphs on n nodes such that every two sets of size n share an edge, having essentially optimal maximum degree n 1?+o(1). Using known and new reductions from these graphs, we explicitly construct: 1. A k round sorting algorithm using n 1+1=k+o(1) comparisons. 2. A k round selection algorithm using n 1+1=(2 k ?1)+o(1) comparisons. 3. A depth 2 superconcentrator of size n 1+o(1). 4. 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 (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 an asymptotically optimal number of random bits from a defective random source using a small additional number of truly random bits.