1
COMPLEXITY THEORY
CSci 5403
LECTURE XXIV:
COMMUNICATION COMPLEXITY
x
2
{0,1}
n
y
2
{0,1}
n
COMMUNICATION
COMPLEXITY
ƒ(x,y) ?
How many bits do Alice and Bob need to send?
Example.
ƒ(x,y) =
∑
i
x
i
+
∑
i
y
i
mod 2 can be
computed with two bits:
∑
i
x
i
mod 2
∑
i
x
i
+
∑
i
y
i
mod 2
Definition.
A k-round two party protocol
∏
to
compute function ƒ: X
×
Y
→
{0,1} is given by
k functions
α
0
,
α
1
,…,
α
k-1
: {0,1}*
→
{0,1}*:
x
2
{0,1}
n
y
2
{0,1}
n
α
0
(x)
α
1
(y,
α
0
)
α
2
(x,
α
1
,
α
0
)
α
k-1
(x,
α
k-2
,…
α
1
,
α
0
) = ƒ(x, y)
The transcript, T
∏
(x,y), is the messages
α
0
,…,
α
k-1
.
ộ
The communication complexity of
∏
is
C(
∏
) = max
x,y
∈
{0,1}
ⁿ
|T
∏
(x,y)|
The complexity of ƒ is C(ƒ) = min
∏
C(
∏
).
Example.
ƒ
<
(x,y): x < y?
ƒ
⊕
(x,y):
∑
i
x
i
+y
i
mod 2
ƒ
MAJ
(x,y):
∑
i
(x
i
+y
i
) > n?
ƒ
DIS
(x,y): x
∩
y = Ø?
ƒ
(x,y):
∑
i
x
i
y
i
mod 2
ƒ
SAT
(
ϕ
,
ψ
):
∃
x,y.
ϕ
(x)
∧
ψ
(y)
ƒ
=
(x,y): x
≟
y
C(ƒ
=
)
≤
n+1
C(ƒ
<
)
≤
n+1
C(ƒ
⊕
)
≤
2
C(ƒ
MAJ
)
≤
log n
C(ƒ
DIS
)
≤
n+1
C(ƒ
)
≤
n+1
C(ƒ
SAT
)
≤
2
Theorem.
∀
ƒ. C(ƒ)
≤
n+1.