Extractors and lower bounds for locally samplable sources
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
On beating the hybrid argument
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Extractors and Lower Bounds for Locally Samplable Sources
ACM Transactions on Computation Theory (TOCT)
The Complexity of Distributions
SIAM Journal on Computing
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Complexity theory typically studies the complexity of computing a function $h(x) : \zo^m \to \zo^n$ of a given input $x$. We advocate the study of the complexity of generating the distribution $h(x)$ for uniform $x$, given random bits. Our main results are: (1) Any function $f : \zo^\ell \to \zon$ such that (i) each output bit $f_i$ depends on $o(\log n)$ input bits, and (ii) $\ell \le \log_2 \binom{n}{\alpha n} + n^{0.99}$, has output distribution $f(U)$ at statistical distance $\ge 1 - 1/n^{0.49}$ from the uniform distribution over $n$-bit strings of hamming weight $\alpha n$. We also prove lower bounds for generating $(X,b(X))$ for boolean $b$, and in the case in which each bit $f_i$ is a small-depth decision tree. These lower bounds seem to be the first of their kind, the proofs use anti-concentration results for the sum of random variables. (2) Lower bounds for generating distributions imply succinct data structures lower bounds. As a corollary of (1), we obtain the first lower bound for the membership problem of representing a set $S \subseteq [n]$ of size $\alpha n$, in the case where $1/\alpha$ is a power of $2$: If queries ``$i \in S$?'' are answered by non-adaptively probing $o(\log n)$ bits, then the representation uses $\ge \log_2 \binom{n}{\alpha n} + \Omega(\log n)$ bits. (3) Upper bounds complementing the bounds in (1) for various settings of parameters. (4) Uniform randomized $\acz$ circuits of $\poly(n)$ size and depth $d = O(1)$ with error $\e$ can be simulated by uniform randomized $\acz$ circuits of $\poly(n)$ size and depth $d+1$ with error $\e + o(1)$ using $\le (\log n)^{O( \log \log n)}$ random bits. Previous derandomizations [Ajtai and Wigderson '85, Nisan '91] increase the depth by a constant factor, or else have poor seed length.