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Unformatted text preview: 6.896 Sublinear Time Algorithms February 6, 2007 Lecture 1 Lecturer: Ronitt Rubinfeld Scribe: Chih-yu Chao 1 Testing Global Properties of Distributions We will begin by considering how to test properties of distributions over large domains. We assume that the only access we have to the distributions is via samples of the distribution, and we will focus on the number of samples required in order to answer our questions about the distribution. We will begin by studying the sample complexity of determining whether two distributions are identical, or far apart according to some measure of distance. 1.1 Closeness Figure 1 : The black box that outputs the distribution of p Here, we will first define the closeness of distributions. Suppose we are given a distribution p in domain D , where | D | = n , and probabilities ( p 1 , . . . p n ), such that p x = Pr[ p outputs x ] unknown to the algorithm, x D p x = 1. We will consider algorithms that are given oracle access to a independent identical distribution p (see Figure 1) each time we press the button on the black box, we get a sample i with probability p i . We would like to consider the L 1 distance between two distributions: Definition 1 ( L 1 distance) | p q | 1 = X x | p x q x | Another common measure is the L 2 distance, defined as follows....
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