Algorithms and Lower Bounds for Comparator Circuits from Shrinkage
In 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-...
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| Veröffentlicht in: | Algorithmica Jg. 85; H. 7; S. 2131 - 2155 |
|---|---|
| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
New York
Springer US
01.07.2023
Springer Nature B.V |
| Schlagworte: | |
| ISSN: | 0178-4617, 1432-0541 |
| Online-Zugang: | Volltext |
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| Zusammenfassung: | In 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
)
with
k
⩾
log
n
, there exists a polynomial-time computable function
f
k
on
n
bits such that, for every comparator circuit
C
with at most
n
1.5
/
O
k
·
log
n
gates, we have
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
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
k
·
log
n
gates, in time
2
n
-
k
·
poly
(
n
)
, for any
k
⩽
n
/
4
. The running time is non-trivial (i.e.,
2
n
/
n
ω
(
1
)
) when
k
=
ω
(
log
n
)
.
Pseudorandom Generators and
MCSP
L
ower Bounds There is a pseudorandom generator of seed length
s
2
/
3
+
o
(
1
)
that fools comparator circuits with
s
gates. Also, using this PRG, we obtain an
n
1.5
-
o
(
1
)
lower bound for
MCSP
against comparator circuits. |
|---|---|
| Bibliographie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-022-01091-y |