The NOF Multiparty Communication Complexity of Composed Functions
We study the k -party “number on the forehead” communication complexity of composed functions f ∘ g → , where f : { 0 , 1 } n → { ± 1 } , g → = ( g 1 , … , g n ) , g i : { 0 , 1 } k → { 0 , 1 } and for ( x 1 , … , x k ) ∈ ( { 0 , 1 } n ) k , f ∘ g → ( x 1 , … , x k ) = f ( … , g i ( x 1 , i , … , x...
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| Vydáno v: | Computational complexity Ročník 24; číslo 3; s. 645 - 694 |
|---|---|
| Hlavní autoři: | , , , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Basel
Springer Basel
01.09.2015
|
| Témata: | |
| ISSN: | 1016-3328, 1420-8954 |
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| Abstract | We study the
k
-party “number on the forehead” communication complexity of composed functions
f
∘
g
→
, where
f
:
{
0
,
1
}
n
→
{
±
1
}
,
g
→
=
(
g
1
,
…
,
g
n
)
,
g
i
:
{
0
,
1
}
k
→
{
0
,
1
}
and for
(
x
1
,
…
,
x
k
)
∈
(
{
0
,
1
}
n
)
k
,
f
∘
g
→
(
x
1
,
…
,
x
k
)
=
f
(
…
,
g
i
(
x
1
,
i
,
…
,
x
k
,
i
)
,
…
)
. When
g
→
=
(
g
,
g
,
…
,
g
)
, we denote
f
∘
g
→
by
f
∘
g
. We show that there is an
O
(
log
3
n
)
cost simultaneous protocol for
SYM
∘
g
when
k
> 1 + log
n
,
SYM
is any symmetric function and
g
is any function. When
k
> 1 + 2 log
n
, our simultaneous protocol applies to
SYM
∘
g
→
with
g
→
being a vector of
n
arbitrary functions. We also get a non-simultaneous protocol for
SYM
∘
g
→
of cost
O
(
n
/
2
k
·
log
n
+
k
log
n
)
for any
k
≥ 2. In the setting of
k
≤ 1 + log
n
, we study more closely functions of the form
MAJORITY
∘
g
,
MOD
m
∘
g
and
NOR
∘
g
, where the latter two are generalizations of the well-known and studied functions generalized inner product and disjointness, respectively. We characterize the communication complexity of these functions with respect to the choice of
g
. In doing so, we answer a question posed by Babai et al. (SIAM J Comput 33:137–166,
2003
) and determine the communication complexity of
MAJORITY
◦
QCSB
k
, where
QCSB
k
is the “quadratic character of the sum of the bits” function.
In the second part of our paper, we utilize the connection between the ‘number on the forehead’ model and Ramsey theory to construct a large set without a
k
-dimensional corner (
k
-dimensional generalization of a
k
-term arithmetic progression) in
(
F
2
n
)
k
, thereby obtaining the first non-trivial bound on the corresponding Ramsey number. Furthermore, we give an explicit coloring of [
N
] × [
N
] without a monochromatic two-dimensional corner and use this to obtain an explicit three-party protocol of cost
O
(
n
)
for the EXACT
N
function. For
x
1
,
x
2
,
x
3
n
-bit integers, EXACT
N
(
x
1
,
x
2
,
x
3
) = −1 iff
x
1
+
x
2
+
x
3
=
N
. |
|---|---|
| AbstractList | We study the
k
-party “number on the forehead” communication complexity of composed functions
f
∘
g
→
, where
f
:
{
0
,
1
}
n
→
{
±
1
}
,
g
→
=
(
g
1
,
…
,
g
n
)
,
g
i
:
{
0
,
1
}
k
→
{
0
,
1
}
and for
(
x
1
,
…
,
x
k
)
∈
(
{
0
,
1
}
n
)
k
,
f
∘
g
→
(
x
1
,
…
,
x
k
)
=
f
(
…
,
g
i
(
x
1
,
i
,
…
,
x
k
,
i
)
,
…
)
. When
g
→
=
(
g
,
g
,
…
,
g
)
, we denote
f
∘
g
→
by
f
∘
g
. We show that there is an
O
(
log
3
n
)
cost simultaneous protocol for
SYM
∘
g
when
k
> 1 + log
n
,
SYM
is any symmetric function and
g
is any function. When
k
> 1 + 2 log
n
, our simultaneous protocol applies to
SYM
∘
g
→
with
g
→
being a vector of
n
arbitrary functions. We also get a non-simultaneous protocol for
SYM
∘
g
→
of cost
O
(
n
/
2
k
·
log
n
+
k
log
n
)
for any
k
≥ 2. In the setting of
k
≤ 1 + log
n
, we study more closely functions of the form
MAJORITY
∘
g
,
MOD
m
∘
g
and
NOR
∘
g
, where the latter two are generalizations of the well-known and studied functions generalized inner product and disjointness, respectively. We characterize the communication complexity of these functions with respect to the choice of
g
. In doing so, we answer a question posed by Babai et al. (SIAM J Comput 33:137–166,
2003
) and determine the communication complexity of
MAJORITY
◦
QCSB
k
, where
QCSB
k
is the “quadratic character of the sum of the bits” function.
In the second part of our paper, we utilize the connection between the ‘number on the forehead’ model and Ramsey theory to construct a large set without a
k
-dimensional corner (
k
-dimensional generalization of a
k
-term arithmetic progression) in
(
F
2
n
)
k
, thereby obtaining the first non-trivial bound on the corresponding Ramsey number. Furthermore, we give an explicit coloring of [
N
] × [
N
] without a monochromatic two-dimensional corner and use this to obtain an explicit three-party protocol of cost
O
(
n
)
for the EXACT
N
function. For
x
1
,
x
2
,
x
3
n
-bit integers, EXACT
N
(
x
1
,
x
2
,
x
3
) = −1 iff
x
1
+
x
2
+
x
3
=
N
. |
| Author | Nguyen, Phuong Fawzi, Omar Ada, Anil Chattopadhyay, Arkadev |
| Author_xml | – sequence: 1 givenname: Anil surname: Ada fullname: Ada, Anil organization: School of Computer Science, McGill University – sequence: 2 givenname: Arkadev surname: Chattopadhyay fullname: Chattopadhyay, Arkadev email: arkadev.c@tifr.res.in organization: School of Technology and Computer Science, Tata Institute of Fundamental Research – sequence: 3 givenname: Omar surname: Fawzi fullname: Fawzi, Omar organization: Institute for Theoretical Physics, ETH – sequence: 4 givenname: Phuong surname: Nguyen fullname: Nguyen, Phuong organization: Dép. d’informatique et de recherche opérationnelle, Université de Montréal |
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| Cites_doi | 10.1006/inco.1994.1051 10.1007/s00037-009-0276-2 10.1070/IM2003v067n01ABEH000422 10.1137/0406009 10.1016/j.jet.2004.10.007 10.1007/PL00001602 10.1073/pnas.32.12.331 10.1007/PL00001592 10.1016/0022-0000(92)90047-M 10.1006/jcss.1997.1545 10.1007/BF01263423 10.1007/BF02790016 10.1137/0222016 10.1137/S0097539702405620 10.4007/annals.2011.174.1.20 10.1007/BF01206318 10.1007/978-3-540-70918-3_43 10.1145/2213977.2214026 10.1016/S0065-2458(08)60342-3 10.1137/0217015 10.1145/28395.28404 10.4007/annals.2007.166.897 10.1070/IM2006v070n02ABEH002316 10.4086/toc.2008.v004a007 10.1017/S0963548303005790 10.1007/s11856-011-0061-1 10.1137/S0097539700375944 10.1006/jcss.1998.1577 10.1007/3-540-59042-0_88 10.1007/11821069_13 10.1109/FOCS.2009.12 10.1007/s000390050105 10.37236/546 10.1109/CCC.2009.24 10.1109/CCC.2011.31 10.26421/QIC9.3-4-5 10.1007/s00039-001-0332-9 10.1145/800061.808737 10.1017/S0024611506015991 10.1006/jcss.1995.1069 10.1112/jlms/s1-28.1.104 10.1112/blms/bdq018 10.1145/800135.804414 10.1137/060654645 10.1145/1015330.1015351 10.4086/toc.2010.v006a009 |
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| Keywords | Number on the forehead model Ramsey theory 05D10 communication complexity 68Q17 68Q05 |
| Language | English |
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| PublicationTitle | Computational complexity |
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| Snippet | We study the
k
-party “number on the forehead” communication complexity of composed functions
f
∘
g
→
, where
f
:
{
0
,
1
}
n
→
{
±
1
}
,
g
→
=
(
g
1
,
…
,
g
n... |
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| SubjectTerms | Algorithm Analysis and Problem Complexity Computational Mathematics and Numerical Analysis Computer Science |
| Title | The NOF Multiparty Communication Complexity of Composed Functions |
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