The Complexity of Distributions

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.