On the relationship between energy complexity and other boolean function measures
We focus on energy complexity , a Boolean function measure related closely to Boolean circuit design. Given a circuit C over the standard basis { ∨ 2 , ∧ 2 , ¬ } , the energy complexity of C , denoted by EC ( C ) , is the maximum number of its activated inner gates over all inputs. The energy comple...
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| Vydáno v: | Journal of combinatorial optimization Ročník 43; číslo 5; s. 1470 - 1492 |
|---|---|
| Hlavní autoři: | , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer US
01.07.2022
Springer Nature B.V |
| Témata: | |
| ISSN: | 1382-6905, 1573-2886 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | We focus on
energy complexity
, a Boolean function measure related closely to Boolean circuit design. Given a circuit
C
over the standard basis
{
∨
2
,
∧
2
,
¬
}
, the energy complexity of
C
, denoted by
EC
(
C
)
, is the maximum number of its activated inner gates over all inputs. The energy complexity of a Boolean function
f
, denoted by
EC
(
f
)
, is the minimum of
EC
(
C
)
over all circuits
C
computing
f
. Recently, K. Dinesh et al. (International computing and combinatorics conference, Springer, Berlin, 738–750, 2018) gave
EC
(
f
)
an upper bound by the decision tree complexity,
EC
(
f
)
=
O
(
D
(
f
)
3
)
. On the input size
n
, They also showed that
EC
(
f
)
is at most
3
n
-
1
. For the lower bound side, they showed that
EC
(
f
)
≥
1
3
psens
(
f
)
, where
psens
(
f
)
is called
positive sensitivity
. A remained open problem is whether the energy complexity of a Boolean function has a polynomial relationship with its decision tree complexity.
Our results for energy complexity can be listed below.
For the lower bound, we prove the equation that
EC
(
f
)
=
Ω
(
D
(
f
)
)
, which answers the above open problem.
For upper bounds,
EC
(
f
)
≤
min
{
1
2
D
(
f
)
2
+
O
(
D
(
f
)
)
,
n
+
2
D
(
f
)
-
2
}
holds.
For non-degenerated functions, we also provide another lower bound
EC
(
f
)
=
Ω
(
log
n
)
where
n
is the input size.
We also discuss the energy complexity of two specific function classes,
OR
functions and
ADDRESS
functions, which implies the tightness of our two lower bounds respectively. In addition, the former one answers another open question in Dinesh et al. (International computing and combinatorics conference, Springer, Berlin, 738–750, 2018) asking for non-trivial lower bound for energy complexity of
OR
functions. |
|---|---|
| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 1382-6905 1573-2886 |
| DOI: | 10.1007/s10878-020-00689-8 |