{"id":161385,"date":"2011-05-01T00:00:00","date_gmt":"2011-05-01T00:00:00","guid":{"rendered":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/msr-research-item\/pseudorandom-generators-for-combinatorial-shapes\/"},"modified":"2018-10-16T21:46:16","modified_gmt":"2018-10-17T04:46:16","slug":"pseudorandom-generators-for-combinatorial-shapes","status":"publish","type":"msr-research-item","link":"https:\/\/newed.any0.dpdns.org\/en-us\/research\/publication\/pseudorandom-generators-for-combinatorial-shapes\/","title":{"rendered":"Pseudorandom Generators for Combinatorial Shapes"},"content":{"rendered":"<div class=\"asset-content\">\n<p>We construct pseudorandom generators for combinatorial shapes, which substantially generalize combinatorial rectangles, -biased spaces, 0\/1 halfspaces, and 0\/1 modular sums. Our generator uses seed length O(logm + log n + log<sup>2<\/sup>(1\/eps)) to get error eps. When m = 2, this gives the fifirst generator of seed length O(log n) which fools all weight-based tests, meaning that the distribution of the weight of any subset is &#8220;-close to the appropriate binomial distribution in statistical distance. Along the way, we give a generator for combinatorial rectangles with seed length O(log<sup>1.5<\/sup> n) and error 1\/poly(n), matching Lu&#8217;s bound [ICALP 1998]. For our proof we give a simple lemma which allows us to convert closeness in Kolmogorov (cdf) distance to closeness in statistical distance. As a corollary of our technique, we give an alternative proof of a powerful variant of the classical central limit theorem showing convergence in statistical distance, instead of the usual Kolmogorov distance.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>We construct pseudorandom generators for combinatorial shapes, which substantially generalize combinatorial rectangles, -biased spaces, 0\/1 halfspaces, and 0\/1 modular sums. Our generator uses seed length O(logm + log n + log2(1\/eps)) to get error eps. When m = 2, this gives the fifirst generator of seed length O(log n) which fools all weight-based tests, meaning 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