n
and 0 < δ < 1, we construct graphs on n nodes such that every two sets of size share an edge, having essentially optimal maximum degree . Using known and new reductions from these graphs, we derive new explicit constructions of:
1. A k round sorting algorithm using comparisons.
2. A k round selection algorithm using comparisons.
3. A depth 2 superconcentrator of size .
4. A depth k wide-sense nonblocking generalized connector of size .
All of these results improve on previous constructions by factors of , and are optimal to within factors of . 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.
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Received: June 13, 1995/Revised: Revised October 23, 1998
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Wigderson, A., Zuckerman, D. Expanders That Beat the Eigenvalue Bound: Explicit Construction and Applications. Combinatorica 19, 125–138 (1999). https://doi.org/10.1007/s004930050049
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DOI: https://doi.org/10.1007/s004930050049