Disperser Graphs, Deterministic Amplification and Imperfect Random Sources

We use a certain type of expanding bipartite graphs, called disperser graphs, to design procedures for sampling a finite number of elements from a finite set, with the property that the probability of hitting any sufficiently large subset is high. The advantages of these procedures are: 1. They require a relatively small number of random bits. 2. They are robust with respect to the quality of the random bits used in the sampling. Using these sampling procedures to sample random inputs of polynomial time probabilistic algorithms, we can simulate the performance of some probabilistic algorithms with less random bits or with low quality random bits. We obtain the following results: 1. The error probability of an RP or BPP algorithm that operates with a constant error bound and requires n random bits, can be made exponentially small (i.e. 2−n), with only (3 + )n random bits, as opposed to standard amplification techniques that require Ω(n) random bits for the same task. This result is nearly optimal, since the information theoretic lower bound on the number of bits required for such an amplification is 2n. 2. It is shown that the output of any random source whose Renyi entropy rate exceeds 1 2 ( 4 ), can be used to simulate RP (BPP) algorithms. This is far from the information theoretic lower bound which is mμ−1, where m is the number of bits and 0 < μ < 1 is any constant. We show that the lower bound can be achieved for a specific class of random sources called oblivious bit fixing sources.