Complexity of Distributions and Average-Case Hardness

We address a natural question in average-case complexity: does there exist a language L such that for all easy distributions D the distributional problem (L,D) is easy on the average while there exists some more hard distribution D′ such that (L,D′) is hard on the average? We consider two complexity measures of distributions: complexity of sampling and complexity of computing the distribution function. The most interesting measure is the complexity of sampling. We prove that for every 0 < a < b there exists a language L, an ensemble of distributions D samplable in nlog b n steps and a linear-time algorithm A such that for every ensemble of distribution F that samplable in nlog a n steps, A correctly decides L on all inputs from {0, 1}n except of a set that has infinitely small F -measure, and for every algorithm B there are infinitely many n such that the set of all elements of {0, 1}n for which B correctly decides L has infinitely small D-measure. In case of complexity of computing the distribution function we prove the following tight result: for every a > 0 there exists a language L, an ensemble of polynomial-time computable distributions D, and a linear-time algorithm A such that for every computable in na steps ensemble of distributions F , A correctly decides L on all inputs from {0, 1}n except a set that has F -measure at most 2−n/2, and for every algorithm B there are infinitely many n such that the set of all elements of {0, 1}n for which B correctly decides L has D-measure at most 2−n+1.