Choosing, Agreeing, and Eliminating in Communication Complexity
We consider several questions inspired by the direct-sum problem in (two-party) communication complexity. In all questions, there are k fixed Boolean functions f 1 ,…, f k and each of Alice and Bob has k inputs, x 1 ,…, x k and y 1 ,…, y k , respectively. In the eliminate problem, Alice and Bob shou...
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| Published in: | Computational complexity 2014-03, Vol.23 (1), p.1-42 |
|---|---|
| Main Authors: | , , , |
| Format: | Article |
| Language: | eng |
| Subjects: | |
| Online Access: | Get full text |
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| Summary: | We consider several questions inspired by the direct-sum problem in (two-party) communication complexity. In all questions, there are
k
fixed Boolean functions
f
1
,…,
f
k
and each of Alice and Bob has
k
inputs,
x
1
,…,
x
k
and
y
1
,…,
y
k
, respectively. In the
eliminate
problem, Alice and Bob should output a vector σ
1
,…,σ
k
such that
f
i
(
x
i
,
y
i
) ≠ σ
i
for at least one
i
(i.e., their goal is to eliminate one of the 2
k
output vectors); in the
choose
problem, Alice and Bob should return (
i
,
f
i
(
x
i
,
y
i
)), for some
i
(i.e., they choose one instance to solve), and in the
agree
problem they should return
f
i
(
x
i
,
y
i
), for some
i
(i.e., if all the
k
Boolean values agree then this must be the output). The question, in each of the three cases, is whether one can do better than solving one (say, the first) instance. We study these three problems and prove various positive and negative results. In particular, we prove that the randomized communication complexity of eliminate, of
k
instances of the same function
f
, is characterized by the randomized communication complexity of solving one instance of
f
. |
|---|---|
| ISSN: | 1016-3328 1420-8954 |