6.3 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Communication complexity | 6/8 | https://en.wikipedia.org/wiki/Communication_complexity | reference | science, encyclopedia | 2026-05-05T14:40:16.151030+00:00 | kb-cron |
== Unbounded-error communication complexity == In the unbounded-error setting, Alice and Bob have access to a private coin and their own inputs
(
x
,
y
)
{\displaystyle (x,y)}
. In this setting, Alice succeeds if she responds with the correct value of
f
(
x
,
y
)
{\displaystyle f(x,y)}
with probability strictly greater than 1/2. In other words, if Alice's responses have any non-zero correlation to the true value of
f
(
x
,
y
)
{\displaystyle f(x,y)}
, then the protocol is considered valid. Note that the requirement that the coin is private is essential. In particular, if the number of public bits shared between Alice and Bob are not counted against the communication complexity, it is easy to argue that computing any function has
O
(
1
)
{\displaystyle O(1)}
communication complexity. On the other hand, both models are equivalent if the number of public bits used by Alice and Bob is counted against the protocol's total communication. Though subtle, lower bounds on this model are extremely strong. More specifically, it is clear that any bound on problems of this class immediately imply equivalent bounds on problems in the deterministic model and the private and public coin models, but such bounds also hold immediately for nondeterministic communication models and quantum communication models. Forster was the first to prove explicit lower bounds for this class, showing that computing the inner product
⟨
x
,
y
⟩
{\displaystyle \langle x,y\rangle }
requires at least
Ω
(
n
)
{\displaystyle \Omega (n)}
bits of communication, though an earlier result of Alon, Frankl, and Rödl proved that the communication complexity for almost all Boolean functions
f
:
{
0
,
1
}
n
×
{
0
,
1
}
n
→
{
0
,
1
}
{\displaystyle f:\{0,1\}^{n}\times \{0,1\}^{n}\to \{0,1\}}
is
Ω
(
n
)
{\displaystyle \Omega (n)}
.
== Lifting == Lifting is a general technique in complexity theory in which a lower bound on a simple measure of complexity is "lifted" to a lower bound on a more difficult measure. This technique was pioneered in the context of communication complexity by Raz and McKenzie, who proved the first query-to-communication lifting theorem, and used the result to separate the monotone NC hierarchy. Given a function
f
:
{
0
,
1
}
n
→
{
0
,
1
}
{\displaystyle f\colon \{0,1\}^{n}\to \{0,1\}}
and a gadget
g
:
{
0
,
1
}
a
×
{
0
,
1
}
b
→
{
0
,
1
}
{\displaystyle g\colon \{0,1\}^{a}\times \{0,1\}^{b}\to \{0,1\}}
, their composition
f
∘
g
:
{
0
,
1
}
n
a
×
{
0
,
1
}
n
b
→
{
0
,
1
}
{\displaystyle f\circ g\colon \{0,1\}^{na}\times \{0,1\}^{nb}\to \{0,1\}}
is defined as follows:
(
f
∘
g
)
(
x
,
y
)
=
f
(
g
(
x
1
,
1
⋯
x
1
,
a
,
y
1
,
1
⋯
y
1
,
b
)
,
…
,
g
(
x
n
,
1
⋯
x
n
,
a
,
y
n
,
1
⋯
y
n
,
b
)
)
.
{\displaystyle (f\circ g)(x,y)=f(g(x_{1,1}\cdots x_{1,a},y_{1,1}\cdots y_{1,b}),\dots ,g(x_{n,1}\cdots x_{n,a},y_{n,1}\cdots y_{n,b})).}
In words,
x
{\displaystyle x}
is partitioned into
n
{\displaystyle n}
blocks of length
a
{\displaystyle a}
, and
y
{\displaystyle y}
is partitioned into
n
{\displaystyle n}
blocks of length
b
{\displaystyle b}
. The gadget is applied
n
{\displaystyle n}
times on the blocks, and the outputs are fed into
f
{\displaystyle f}
. Diagrammatically: