Block sensitivity of minterm-transitive functions
Boolean functions with a high degree of symmetry are interesting from a complexity theory perspective: extensive research has shown that these functions, if nonconstant, must have high complexity according to various measures. In a recent work of this type, Sun (2007) [9] gave lower bounds on the bl...
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| Vydáno v: | Theoretical computer science Ročník 412; číslo 41; s. 5796 - 5801 |
|---|---|
| Hlavní autor: | |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Oxford
Elsevier B.V
23.09.2011
Elsevier |
| Témata: | |
| ISSN: | 0304-3975, 1879-2294 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | Boolean functions with a high degree of symmetry are interesting from a complexity theory perspective: extensive research has shown that these functions, if nonconstant, must have high complexity according to various measures.
In a recent work of this type, Sun (2007)
[9] gave lower bounds on the block sensitivity of nonconstant Boolean functions invariant under a transitive permutation group. Sun showed that all such functions satisfy
b
s
(
f
)
=
Ω
(
N
1
/
3
)
. He also showed that there exists such a function for which
b
s
(
f
)
=
O
(
N
3
/
7
ln
N
)
. His example belongs to a subclass of transitively invariant functions called “minterm-transitive” functions, defined by Chakraborty (2005)
[3].
We extend these results in two ways. First, we show that nonconstant minterm-transitive functions satisfy
b
s
(
f
)
=
Ω
(
N
3
/
7
)
. Thus, Sun’s example has nearly minimal block sensitivity for this subclass. Second, we improve Sun’s example: we exhibit a minterm-transitive function for which
b
s
(
f
)
=
O
(
N
3
/
7
ln
1
/
7
N
)
. |
|---|---|
| Bibliografie: | ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23 |
| ISSN: | 0304-3975 1879-2294 |
| DOI: | 10.1016/j.tcs.2011.06.025 |