1
Introduction
In this report, we study the multiparty communication complexity problem of the multiparty equality
function (MEQ):
EQ
(
x
1
,
· · ·
, x
n
) =
{
0
if
x
1
=
· · ·
=
x
n
1
otherwise
.
(1)
The input vector
x
= (
x
1
,
· · ·
, x
n
) is distributed among
n
≥
2 nodes, with
x
i
known to node
i
, where
x
i
is chosen from the set
{
1
,
· · ·
, M
}
, for some integer
M >
0.
1.1
Communication Complexity
The notion of communication complexity (CC) was introduced by Yao in 1979 [2], who investigated
the following problem involving two separated parties (Alice and Bob) want to mutually compute a
Boolean function that is defined on pairs of inputs. Formally, let
f
:
X
×
Y
7→ {
0
,
1
}
be a Boolean
function. The communication problem for
f
is the following two-party game:
Alice receives
x
∈
X
and Bob receives
y
∈
Y
, and the goal is for them to compute
f
(
x, y
),
collaboratively.
Alice and Bob have unlimited computational power and a full description of
f
, but
they do not know each other’s input. They determine the output value by exchanging messages. The
computation ends when either Alice or Bob has enough information to determine
f
(
x, y
), and sends a
special symbol “halt” to the other party.
A protocol
P
for computing
f
is an algorithm, according to which Alice and Bob send binary
messages to each other. A protocol proceeds in rounds. In every round, the protocol specifies whose
turn it is to send a message. Each party in his/her turn sends one bit that may depend on his/her
input and the previous messages he/she has received. A correct protocol for
f
should terminate for
every input pair (
x, y
)
∈
X
×
Y
, when either Alice or Bob knows
f
(
x, y
).
The communication complexity of a protocol
P
is the number of bits exchanged for the worst case
input pair. The communication complexity of a Boolean function
f
:
X
×
Y
7→ {
0
,
1
}
, is that of the
protocols for
f
with the least complexity.
1.2
Multiparty Communication Complexity
There is more than one way to generalize communication complexity to a multiparty setting. The most
commonly used model is the “number on the forehead” model introduced in [1].
Formally, there is
some function
f
: Π
n
i
=1
X
i
7→ {
0
,
1
}
, and the input is (
x
1
, x
2
,
· · ·
, x
n
) where each
x
i
∈
X
i
.
The
i
-th
party can see all the
x
j
such that
j
̸
=
i
. As in the 2-party case, the
n
players have an agreed-upon
protocol for communication, and all this communication is posted on a “public blackboard”. At the