2006 | OriginalPaper | Chapter
Randomness-Efficient Sampling Within NC 1
Author : Alexander Healy
Published in: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Publisher: Springer Berlin Heidelberg
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We construct a randomness-efficient
averaging sampler
that is computable by uniform constant-depth circuits with parity gates (i.e., in uniform
AC
0
[⊕]). Our sampler matches the parameters achieved by random walks on constant-degree expander graphs, allowing us to apply a variety expander-based techniques within
NC
1
. For example, we obtain the following results:
Randomness-efficient error-reduction for uniform probabilistic
NC
1
,
TC
0
,
AC
0
[⊕] and
AC
0
: 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
AC
0
[⊕]: There is a 1/2
Ω(
n
)
-biased generator
$G:{0, 1}^{O(n)} \to {0, 1}^{2^n}$
for which poly(
n
)-size uniform
AC
0
[⊕] circuits can compute
G
(
s
)
i
given (
s
,
i
) ∈0, 1
O
(
n
)
×0, 1
n
. This resolves a question raised by Gutfreund & Viola (
Random 2004
).
uniform BP ·
AC
0
⊆ uniform
AC
0
/
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
).