Author:
Cavalar Bruno P.,Lu Zhenjian
Abstract
AbstractIn this paper, we initiate the study of average-case complexity and circuit analysis algorithms for comparator circuits. Departing from previous approaches, we exploit the technique of shrinkage under random restrictions to obtain a variety of new results for this model. Among them, we show
Average-case Lower Bounds For every $$k = k(n)$$
k
=
k
(
n
)
with $$k \geqslant \log n$$
k
⩾
log
n
, there exists a polynomial-time computable function $$f_k$$
f
k
on n bits such that, for every comparator circuit C with at most $$n^{1.5}/O\!\left( k\cdot \sqrt{\log n}\right) $$
n
1.5
/
O
k
·
log
n
gates, we have $$\begin{aligned} \mathop {{{\,\mathrm{\textbf{Pr}}\,}}}\limits _{x\in \left\{ 0,1\right\} ^n}\left[ C(x)=f_k(x)\right] \leqslant \frac{1}{2} + \frac{1}{2^{\Omega (k)}}. \end{aligned}$$
Pr
x
∈
0
,
1
n
C
(
x
)
=
f
k
(
x
)
⩽
1
2
+
1
2
Ω
(
k
)
.
This average-case lower bound matches the worst-case lower bound of Gál and Robere by letting $$k=O\!\left( \log n\right) $$
k
=
O
log
n
.
$$\#$$
#
SAT Algorithms There is an algorithm that counts the number of satisfying assignments of a given comparator circuit with at most $$n^{1.5}/O\!\left( k\cdot \sqrt{\log n}\right) $$
n
1.5
/
O
k
·
log
n
gates, in time $$2^{n-k}\cdot {{\,\textrm{poly}\,}}(n)$$
2
n
-
k
·
poly
(
n
)
, for any $$k\leqslant n/4$$
k
⩽
n
/
4
. The running time is non-trivial (i.e., $$2^n/n^{\omega (1)}$$
2
n
/
n
ω
(
1
)
) when $$k=\omega (\log n)$$
k
=
ω
(
log
n
)
.
Pseudorandom Generators and $$\textsf {MCSP} $$
MCSP
Lower Bounds There is a pseudorandom generator of seed length $$s^{2/3+o(1)}$$
s
2
/
3
+
o
(
1
)
that fools comparator circuits with s gates. Also, using this PRG, we obtain an $$n^{1.5-o(1)}$$
n
1.5
-
o
(
1
)
lower bound for $$\textsf {MCSP} $$
MCSP
against comparator circuits.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computer Science Applications,General Computer Science