## 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 language | English (US) |
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Pages (from-to) | 123-154 |

Number of pages | 32 |

Journal | Journal of the ACM |

Volume | 45 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1998 |

## All Science Journal Classification (ASJC) codes

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

## Keywords

- Algorithms
- F.1.3 [Computaion by Abstract Devices]: Complexity Classes - relations among randomized complexity classes
- G.2.1 [Discrete Mathematics]: Combinatorics - combinatorial algorithms
- G.3 [Probability and Statistics]: probabilistic algorithms