Explicit OR-dispersers with polylogarithmic degree

Michael Saks, Aravind Srinivasan, Shiyu Zhou

Research output: Contribution to journalArticlepeer-review

21 Scopus citations

Abstract

An (N, M, T)-OR-disperser is a bipartite multigraph G = (V, W, E) with |V| = N, and |W| = M, having the following expansion property: any subset of V having at least T vertices has a neighbor set of size at least M/2. For any pair of constants ξ, λ, 1 ≥ ξ > λ ≥ 0, any sufficiently large N, and for any T ≥ 2(log N)ξ, M ≤ 2(log N)λ, we give an explicit elementary construction of an (N, M, T)-OR-disperser such that the out-degree of any vertex in V is at most polylogarithmic in N. Using this with known applications of OR-dispersers yields several results. First, our construction implies that the complexity class Strong-RP defined by Sipser, equals RP. Second, for any fixed η > 0, we give the first polynomial-time simulation of RP algorithms using the output of any "η-minimally random" source. For any integral R > 0, such a source accepts a single request for an R-bit string and generates the string according to a distribution that assigns probability at most 2-Rη to any string. It is minimally random in the sense that any weaker source is insufficient to do a black-box polynomial-time simulation of RP algorithms.

Original languageEnglish (US)
Pages (from-to)123-154
Number of pages32
JournalJournal of the ACM
Volume45
Issue number1
DOIs
StatePublished - Jan 1 1998

All Science Journal Classification (ASJC) codes

  • Software
  • Control and Systems Engineering
  • Information Systems
  • Hardware and Architecture
  • Artificial Intelligence

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