Explicit OR-dispersers with polylogarithmic degree

An (<italic>N, M, T</italic>)-OR-disperser is a bipartite multigraph <italic>G</italic>=(<italic>V, W, E</italic>) with |<italic>V</italic>| = <italic>N</italic>, and |<italic>W</italic>| = <italic>M</italic>, having the following expansion property: any subset of <italic>V</italic> having at least <italic>T</italic> vertices has a neighbor set of size at least <italic>M</italic>/2. For any pair of constants &xgr;, λ, 1 ≥ &xgr; > λ ≥ 0, any sufficiently large <italic>N</italic>, and for any <italic>T</italic> ≥ 2<supscrpt>(log<italic>N</italic>)</supscrpt> <italic>M</italic> ≤ 2<supscrpt>(log <italic>N</italic>)<supscrpt>λ</supscrpt></supscrpt>, we give an explicit elementary construction of an (<italic>N, M, T</italic>)-OR-disperser such that the out-degree of any vertex in <italic>V</italic> is at most polylogarithmic in <italic>N</italic>. 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 <italic>R</italic> > 0, such a source accepts a single request for an <italic>R</italic>-bit string and generates the string according to a distribution that assigns probability at most 2<supscrpt>−R<supscrpt>&eegr;</supscrpt></supscrpt> 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.