@MISC{Watson13thecomputational, author = {Thomas Weir Watson}, title = {The Computational Complexity of Randomness }, year = {2013} }

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Abstract

This dissertation explores the multifaceted interplay between efficient computation andprobability distributions. We organize the aspects of this interplay according to whether the randomness occurs primarily at the level of the problem or the level of the algorithm, and orthogonally according to whether the output is random or the input is random. Part I concerns settings where the problem’s output is random. A sampling problem associates to each input x a probability distribution D(x), and the goal is to output a sample from D(x) (or at least get statistically close) when given x. Although sampling algorithms are fundamental tools in statistical physics, combinatorial optimization, and cryptography, and algorithms for a wide variety of sampling problems have been discovered, there has been comparatively little research viewing sampling throughthelens ofcomputational complexity. We contribute to the understanding of the power and limitations of efficient sampling by proving a time hierarchy theorem which shows, roughly, that “a little more time gives a lot more power to sampling algorithms.” Part II concerns settings where the algorithm’s output is random. Even when the specificationofacomputational problem involves no randomness, onecanstill consider randomized