Randomness-efficient sampling within NC1

• Authors: Alexander Healy
• Venue: APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
• Year: 2006

We construct a randomness-efficient averaging sampler that is computable by uniform constant-depth circuits with parity gates (i.e., in uniform AC0[⊕]). Our sampler matches the parameters achieved by random walks on constant-degree expander graphs, allowing us to apply a variety expander-based techniques within NC1. For example, we obtain the following results: Randomness-efficient error-reduction for uniform probabilistic NC1, TC0, AC0[⊕] and AC0: Any function computable by uniform probabilistic circuits with error 1/3 using r random bits is computable by uniform probabilistic circuits with error δ using r + O(log(1/δ)) random bits. Optimal explicit ε-biased generator in AC0[⊕]: There is a 1/2Ω( n)-biased generator $G:{0, 1}^{O(n)} \to {0, 1}^{2^n}$ for which poly(n)-size uniform AC0[⊕] circuits can compute G(s)i given (s, i) ∈0, 1O( n) ×0, 1n. This resolves a question raised by Gutfreund & Viola (Random 2004). uniform BP ·AC0⊆ uniform AC0/O(n). Our sampler is based on the zig-zag graph product of Reingold, Vadhan and Wigderson (Annals of Math 2002) and as part of our analysis we give an elementary proof of a generalization of Gillman's Chernoff Bound for Expander Walks (FOCS 1994).