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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Communication complexity | 7/8 | https://en.wikipedia.org/wiki/Communication_complexity | reference | science, encyclopedia | 2026-05-05T14:40:16.151030+00:00 | kb-cron |
In this diagram, each of the inputs
x
1
,
…
,
x
n
{\displaystyle \mathbf {x} _{1},\dots ,\mathbf {x} _{n}}
is a bits long, and each of the inputs
y
1
,
…
,
y
n
{\displaystyle \mathbf {y} _{1},\dots ,\mathbf {y} _{n}}
is b bits long. A decision tree of depth
Δ
{\displaystyle \Delta }
for
f
{\displaystyle f}
can be translated to a communication protocol whose cost is
Δ
⋅
D
(
g
)
{\displaystyle \Delta \cdot D(g)}
: each time the tree queries a bit, the corresponding value of
g
{\displaystyle g}
is computed using an optimal protocol for
g
{\displaystyle g}
. Raz and McKenzie showed that this is optimal up to a constant factor when
g
{\displaystyle g}
is the so-called "indexing gadget", in which
x
{\displaystyle x}
has length
c
log
n
{\displaystyle c\log n}
(for a large enough constant c),
y
{\displaystyle y}
has length
n
c
{\displaystyle n^{c}}
, and
g
(
x
,
y
)
{\displaystyle g(x,y)}
is the
x
{\displaystyle x}
-th bit of
y
{\displaystyle y}
. The proof of the Raz–McKenzie lifting theorem uses the method of simulation, in which a protocol for the composed function
f
∘
g
{\displaystyle f\circ g}
is used to generate a decision tree for
f
{\displaystyle f}
. Göös, Pitassi and Watson gave an exposition of the original proof. Since then, several works have proved similar theorems with different gadgets, such as inner product. The smallest gadget that can be handled is the indexing gadget with
c
=
1
+
ϵ
{\displaystyle c=1+\epsilon }
. Göös, Pitassi and Watson extended the Raz–McKenzie technique to randomized protocols. A simple modification of the Raz–McKenzie lifting theorem gives a lower bound of
Δ
⋅
D
(
g
)
{\displaystyle \Delta \cdot D(g)}
on the logarithm of the size of a protocol tree for computing
f
∘
g
{\displaystyle f\circ g}
, where
Δ
{\displaystyle \Delta }
is the depth of the optimal decision tree for
f
{\displaystyle f}
. Garg, Göös, Kamath and Sokolov extended this to the DAG-like setting, and used their result to obtain monotone circuit lower bounds. The same technique has also yielded applications to proof complexity. A different type of lifting is exemplified by Sherstov's pattern matrix method, which gives a lower bound on the quantum communication complexity of
f
∘
g
{\displaystyle f\circ g}
, where g is a modified indexing gadget, in terms of the approximate degree of f. The approximate degree of a Boolean function is the minimal degree of a polynomial that approximates the function on all Boolean points up to an additive error of 1/3. In contrast to the Raz–McKenzie proof, which uses the method of simulation, Sherstov's proof takes a dual witness to the approximate degree of f and gives a lower bound on the quantum query complexity of
f
∘
g
{\displaystyle f\circ g}
using the generalized discrepancy method. The dual witness for the approximate degree of f is a lower bound witness for the approximate degree obtained via LP duality. This dual witness is massaged into other objects constituting data for the generalized discrepancy method. Another example of this approach is the work of Pitassi and Robere, in which an algebraic gap is lifted to a lower bound on Razborov's rank measure. The result is a strongly exponential lower bound on the monotone circuit complexity of an explicit function, obtained via the Karchmer–Wigderson characterization of monotone circuit size in terms of communication complexity.
== Open problems == Considering a 0 or 1 input matrix
M
f
=
[
f
(
x
,
y
)
]
x
,
y
∈
{
0
,
1
}
n
{\displaystyle M_{f}=[f(x,y)]_{x,y\in \{0,1\}^{n}}}
, the minimum number of bits exchanged to compute
f
{\displaystyle f}
deterministically in the worst case,
D
(
f
)
{\displaystyle D(f)}
, is known to be bounded from below by the logarithm of the rank of the matrix
M
f
{\displaystyle M_{f}}
. The log rank conjecture proposes that the communication complexity,
D
(
f
)
{\displaystyle D(f)}
, is bounded from above by a constant power of the logarithm of the rank of
M
f
{\displaystyle M_{f}}
. Since D(f) is bounded from above and below by polynomials of log rank
(
M
f
)
{\displaystyle (M_{f})}
, we can say D(f) is polynomially related to log rank
(
M
f
)
{\displaystyle (M_{f})}
. Since the rank of a matrix is polynomial time computable in the size of the matrix, such an upper bound would allow the matrix's communication complexity to be approximated in polynomial time. Note, however, that the size of the matrix itself is exponential in the size of the input. For a randomized protocol, the number of bits exchanged in the worst case, R(f), was conjectured to be polynomially related to the following formula:
log
min
(
rank
(
M
f
′
)
:
M
f
′
∈
R
2
n
×
2
n
,
(
M
f
−
M
f
′
)
∞
≤
1
/
3
)
.
{\displaystyle \log \min({\textrm {rank}}(M'_{f}):M'_{f}\in \mathbb {R} ^{2^{n}\times 2^{n}},(M_{f}-M'_{f})_{\infty }\leq 1/3).}
Such log rank conjectures are valuable because they reduce the question of a matrix's communication complexity to a question of linearly independent rows (columns) of the matrix. This particular version, called the Log-Approximate-Rank Conjecture, was recently refuted by Chattopadhyay, Mande and Sherif (2019) using a surprisingly simple counter-example. This reveals that the essence of the communication complexity problem, for example in the EQ case above, is figuring out where in the matrix the inputs are, in order to find out if they're equivalent.