Explicit OR-dispersers with polylogarithmic degree

  • Authors:

  • Michael Saks;Aravind Srinivasan;Shiyu Zhou

  • Affiliations:

  • Rutgers Univ., New Brunswick, NJ;National Univ. of Singapore, Singapore;Bell Labs, Murray Hill, NJ

  • Venue:
  • Journal of the ACM (JACM)
  • Year:
  • 1998

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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 &xgr;, &lgr;, 1 ≥ &xgr; &lgr; ≥ 0, any sufficiently large N, and for any T ≥ 2(logN) M ≤ 2(log N)&lgr;, 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 &eegr; 0, we give the first polynomial-time simulation of RP algorithms using the output of any “&eegr;-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−R&eegr; 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.