How to generate cryptographically strong sequences of pseudo-random bits
SIAM Journal on Computing
A fast and simple randomized parallel algorithm for the maximal independent set problem
Journal of Algorithms
Computational limitations of small-depth circuits
Computational limitations of small-depth circuits
One-way functions and circuit complexity
Information and Computation
Random oracles separate PSPACE from the polynomial-time hierarchy
Information Processing Letters
On the power of two-point based sampling
Journal of Complexity
Converting high probability into nearly-constant time—with applications to parallel hashing
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
Fast parallel generation of random permutations
Proceedings of the 18th international colloquium on Automata, languages and programming
Journal of Computer and System Sciences
Delayed path coupling and generating random permutations via distributed stochastic processes
Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms
On the cell probe complexity of membership and perfect hashing
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Low Redundancy in Static Dictionaries with Constant Query Time
SIAM Journal on Computing
SIAM Journal on Computing
The Quantum Communication Complexity of Sampling
SIAM Journal on Computing
On Constructing Parallel Pseudorandom Generators from One-Way Functions
CCC '05 Proceedings of the 20th Annual IEEE Conference on Computational Complexity
On the randomness complexity of efficient sampling
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Random Structures & Algorithms
Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing)
Elements of Information Theory (Wiley Series in Telecommunications and Signal Processing)
Pseudorandom Bits for Constant-Depth Circuits with Few Arbitrary Symmetric Gates
SIAM Journal on Computing
SIAM Journal on Computing
Computational Complexity: A Conceptual Perspective
Computational Complexity: A Conceptual Perspective
Theory and application of trapdoor functions
SFCS '82 Proceedings of the 23rd Annual Symposium on Foundations of Computer Science
The bit extraction problem or t-resilient functions
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
On Pseudorandom Generators with Linear Stretch in NC0
Computational Complexity
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Unbalanced expanders and randomness extractors from Parvaresh--Vardy codes
Journal of the ACM (JACM)
Concentration of Measure for the Analysis of Randomized Algorithms
Concentration of Measure for the Analysis of Randomized Algorithms
Pseudorandom Bit Generators That Fool Modular Sums
APPROX '09 / RANDOM '09 Proceedings of the 12th International Workshop and 13th International Workshop on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Polylogarithmic independence fools AC0 circuits
Journal of the ACM (JACM)
Changing base without losing space
Proceedings of the forty-second ACM symposium on Theory of computing
Fooling Functions of Halfspaces under Product Distributions
CCC '10 Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity
Cell-probe lower bounds for succinct partial sums
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Pseudorandom Bits for Polynomials
SIAM Journal on Computing
The Complexity of Distributions
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
On the Implementation of Huge Random Objects
SIAM Journal on Computing
Bounded-Depth Circuits Cannot Sample Good Codes
CCC '11 Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity
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
Bounded Independence Fools Halfspaces
SIAM Journal on Computing
Extractors for Circuit Sources
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Time hierarchies for sampling distributions
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Verifying proofs in constant depth
ACM Transactions on Computation Theory (TOCT)
Hi-index | 0.00 |
Complexity theory typically studies the complexity of computing a function $h(x) : \{0, 1\}^m \to \{0, 1\}^n$ of a given input $x$. A few works have suggested studying the complexity of generating—or sampling—the distribution $h(x)$ for uniform $x$, given random bits. We further advocate this study, with a new emphasis on lower bounds for restricted computational models. Our main results are the following: (1) Any function $f : \{0, 1\}^\ell \to \{0, 1\}^n$ 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 anticoncentration results for the sum of random variables. (2) Lower bounds for succinct data structures. 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 nonadaptively 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 $\mathrm{AC}^0$ circuits of $\mathrm{poly}(n)$ size and depth $d = O(1)$ with error $\epsilon$ can be simulated by uniform randomized $\mathrm{AC}^0$ circuits of $\mathrm{poly}(n)$ size and depth $d+1$ with error $\epsilon + o(1)$ using $\le (\log n)^{O( \log \log n)}$ random bits. Previous derandomizations [M. Ajtai and A. Wigderson, Adv. Comput. Res., 5 (1989), pp. 199-223], [N. Nisan, Combinatorica, 11 (1991), pp. 63-70] increase the depth by a constant factor, or else have poor seed length.