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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| Vydané v: | Computational complexity Ročník 24; číslo 3; s. 645 - 694 |
|---|---|
| Hlavní autori: | , , , |
| Médium: | Journal Article |
| Jazyk: | English |
| Vydavateľské údaje: |
Basel
Springer Basel
01.09.2015
|
| Predmet: | |
| ISSN: | 1016-3328, 1420-8954 |
| On-line prístup: | Získať plný text |
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| Shrnutí: | 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
. |
|---|---|
| ISSN: | 1016-3328 1420-8954 |
| DOI: | 10.1007/s00037-013-0078-4 |