Explicit Or-dispersers with Polylogarithmic Degree 1
نویسندگان
چکیده
An (N;M;T )-OR-disperser is a bipartite multigraph G = (V;W;E) with jV j = N , and jW j = 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 ; ; 1 > 0, any su ciently large N , and for any T 2(logN) , M 2(logN) , 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 de ned by Sipser, equals RP. Second, for any xed > 0, we give the rst polynomial-time simulation of RP algorithms using the output of any \ -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 to any string. It is minimally random in the sense that any weaker source is insu cient to do a black-box polynomial-time simulation of RP algorithms. { 2 {
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