Expanders that beat the eigenvalue bound: Explicit construction and applications

Avi Wigderson, David Zuckerman

Research output: Contribution to journalArticlepeer-review

56 Scopus citations

Abstract

For every n and 0 < δ < 1, we construct graphs on n nodes such that every two sets of size nδ share an edge, having essentially optimal maximum degree n1-δ+o(1). Using known and new reductions from these graphs, we derive new explicit constructions of: 1. A k round sorting algorithm using n1+1/k+o(1) comparisons. 2. A k round selection algorithm using n1+1/(2k-1)+o(1) comparisons. 3. A depth 2 superconcentrator of size n1+o(1). 4. A depth k wide-sense nonblocking generalized connector of size n1+1/k+o(1). All of these results improve on previous constructions by factors of nΩ(1), and are optimal to within factors of no(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.

Original languageEnglish
Pages (from-to)125-138
Number of pages14
JournalCombinatorica
Volume19
Issue number1
DOIs
StatePublished - 1999

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