Quadratically Tight Relations for Randomized Query Complexity
In this work we investigate the problem of quadratically tightly approximating the randomized query complexity of Boolean functions R ( f ). The certificate complexity C ( f ) is such a complexity measure for the zero-error randomized query complexity R 0 ( f ): C ( f ) ≤ R 0 ( f ) ≤ C ( f ) 2 . In...
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| Vydané v: | Theory of computing systems Ročník 64; číslo 1; s. 101 - 119 |
|---|---|
| Hlavní autori: | , , , , , , |
| Médium: | Journal Article |
| Jazyk: | English |
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New York
Springer US
01.01.2020
Springer Nature B.V Springer Verlag |
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| ISSN: | 1432-4350, 1433-0490 |
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| Abstract | In this work we investigate the problem of quadratically tightly approximating the randomized query complexity of Boolean functions
R
(
f
). The certificate complexity
C
(
f
) is such a complexity measure for the zero-error randomized query complexity
R
0
(
f
):
C
(
f
) ≤
R
0
(
f
) ≤
C
(
f
)
2
. In the first part of the paper we introduce a new complexity measure, expectational certificate complexity
E
C
(
f
), which is also a quadratically tight bound on
R
0
(
f
):
E
C
(
f
) ≤
R
0
(
f
) =
O
(
E
C
(
f
)
2
). For
R
(
f
), we prove that
E
C
2/3
≤
R
(
f
). We then prove that
E
C
(
f
) ≤
C
(
f
) ≤
E
C
(
f
)
2
and show that there is a quadratic separation between the two, thus
E
C
(
f
) gives a tighter upper bound for
R
0
(
f
). The measure is also related to the fractional certificate complexity
F
C
(
f
) as follows:
F
C
(
f
) ≤
E
C
(
f
) =
O
(
F
C
(
f
)
3/2
). This also connects to an open question by Aaronson whether
F
C
(
f
) is a quadratically tight bound for
R
0
(
f
), as
E
C
(
f
) is in fact a relaxation of
F
C
(
f
). In the second part of the work, we investigate whether the corruption bound
c
o
r
r
𝜖
(
f
) quadratically approximates
R
(
f
). By Yao’s theorem, it is enough to prove that the square of the corruption bound upper bounds the distributed query complexity
D
𝜖
μ
(
f
)
for all input distributions
μ
. Here, we show that this statement holds for input distributions in which the various bits of the input are distributed independently. This is a natural and interesting subclass of distributions, and is also in the spirit of the input distributions studied in communication complexity in which the inputs to the two communicating parties are statistically independent. Our result also improves upon a result of Harsha et al. (
2016
), who proved a similar weaker statement. We also note that a similar statement in the communication complexity is open. |
|---|---|
| AbstractList | In this work we investigate the problem of quadratically tightly approximating the randomized query complexity of Boolean functions
R
(
f
). The certificate complexity
C
(
f
) is such a complexity measure for the zero-error randomized query complexity
R
0
(
f
):
C
(
f
) ≤
R
0
(
f
) ≤
C
(
f
)
2
. In the first part of the paper we introduce a new complexity measure, expectational certificate complexity
E
C
(
f
), which is also a quadratically tight bound on
R
0
(
f
):
E
C
(
f
) ≤
R
0
(
f
) =
O
(
E
C
(
f
)
2
). For
R
(
f
), we prove that
E
C
2/3
≤
R
(
f
). We then prove that
E
C
(
f
) ≤
C
(
f
) ≤
E
C
(
f
)
2
and show that there is a quadratic separation between the two, thus
E
C
(
f
) gives a tighter upper bound for
R
0
(
f
). The measure is also related to the fractional certificate complexity
F
C
(
f
) as follows:
F
C
(
f
) ≤
E
C
(
f
) =
O
(
F
C
(
f
)
3/2
). This also connects to an open question by Aaronson whether
F
C
(
f
) is a quadratically tight bound for
R
0
(
f
), as
E
C
(
f
) is in fact a relaxation of
F
C
(
f
). In the second part of the work, we investigate whether the corruption bound
c
o
r
r
𝜖
(
f
) quadratically approximates
R
(
f
). By Yao’s theorem, it is enough to prove that the square of the corruption bound upper bounds the distributed query complexity
D
𝜖
μ
(
f
)
for all input distributions
μ
. Here, we show that this statement holds for input distributions in which the various bits of the input are distributed independently. This is a natural and interesting subclass of distributions, and is also in the spirit of the input distributions studied in communication complexity in which the inputs to the two communicating parties are statistically independent. Our result also improves upon a result of Harsha et al. (
2016
), who proved a similar weaker statement. We also note that a similar statement in the communication complexity is open. In this work we investigate the problem of quadratically tightly approximating the randomized query complexity of Boolean functions R(f). The certificate complexity C(f) is such a complexity measure for the zero-error randomized query complexity R0(f): C(f) ≤R0(f) ≤C(f)2. In the first part of the paper we introduce a new complexity measure, expectational certificate complexity EC(f), which is also a quadratically tight bound on R0(f): EC(f) ≤R0(f) = O(EC(f)2). For R(f), we prove that EC2/3 ≤R(f). We then prove that EC(f) ≤C(f) ≤EC(f)2 and show that there is a quadratic separation between the two, thus EC(f) gives a tighter upper bound for R0(f). The measure is also related to the fractional certificate complexity FC(f) as follows: FC(f) ≤EC(f) = O(FC(f)3/2). This also connects to an open question by Aaronson whether FC(f) is a quadratically tight bound for R0(f), as EC(f) is in fact a relaxation of FC(f). In the second part of the work, we investigate whether the corruption bound corrðoe-(f) quadratically approximates R(f). By Yao’s theorem, it is enough to prove that the square of the corruption bound upper bounds the distributed query complexity Dðoe-μ(f)\(\mathsf {D}^{\mu }_{\epsilon }(f)\) for all input distributions μ. Here, we show that this statement holds for input distributions in which the various bits of the input are distributed independently. This is a natural and interesting subclass of distributions, and is also in the spirit of the input distributions studied in communication complexity in which the inputs to the two communicating parties are statistically independent. Our result also improves upon a result of Harsha et al. (2016), who proved a similar weaker statement. We also note that a similar statement in the communication complexity is open. |
| Author | Jain, Rahul Santha, Miklos Sanyal, Swagato Lee, Troy Kundu, Srijita Klauck, Hartmut Vihrovs, Jevgēnijs |
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| Copyright | Springer Science+Business Media, LLC, part of Springer Nature 2019 Theory of Computing Systems is a copyright of Springer, (2019). All Rights Reserved. Distributed under a Creative Commons Attribution 4.0 International License |
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| Issue | 1 |
| Keywords | Corruption bound Randomized algorithms Certificate complexity Query complexity Fractional block sensitivity |
| Language | English |
| License | Distributed under a Creative Commons Attribution 4.0 International License: http://creativecommons.org/licenses/by/4.0 |
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| References | CR2 Kulkarni, Tal (CR11) 2016; 8 Aaronson (CR1) 2008; 74 Buhrman, de Wolf (CR6) 2002; 288 Gilmer, Saks, Srinivasan (CR7) 2016; 36 CR3 CR5 CR8 CR9 CR13 Tardos (CR14) 1990; 9 CR12 CR10 Beals, Buhrman, Cleve, Mosca, de Wolf (CR4) 2001; 48 9935_CR2 R Beals (9935_CR4) 2001; 48 9935_CR3 J Gilmer (9935_CR7) 2016; 36 9935_CR10 9935_CR12 9935_CR13 S Aaronson (9935_CR1) 2008; 74 9935_CR8 9935_CR9 G Tardos (9935_CR14) 1990; 9 R Kulkarni (9935_CR11) 2016; 8 9935_CR5 H Buhrman (9935_CR6) 2002; 288 |
| References_xml | – volume: 74 start-page: 313 issue: 3 year: 2008 end-page: 322 ident: CR1 article-title: Quantum certificate complexity publication-title: J. Comput. Syst. Sci. doi: 10.1016/j.jcss.2007.06.020 – ident: CR3 – ident: CR2 – volume: 9 start-page: 385 year: 1990 end-page: 392 ident: CR14 article-title: Query complexity or why is it difficult to separate NPA ∩coNPA from PA by a random oracle publication-title: Combinatorica doi: 10.1007/BF02125350 – volume: 8 start-page: 1 year: 2016 end-page: 16 ident: CR11 article-title: On fractional block sensitivity publication-title: Chicago Journal Of Theoretical Computer Science doi: 10.4086/cjtcs.2016.008 – ident: CR12 – ident: CR13 – ident: CR10 – ident: CR9 – volume: 36 start-page: 265 issue: 3 year: 2016 end-page: 311 ident: CR7 article-title: Composition limits and separating examples for some Boolean function complexity measures publication-title: Combinatorica doi: 10.1007/s00493-014-3189-x – volume: 288 start-page: 21 issue: 1 year: 2002 end-page: 43 ident: CR6 article-title: Complexity measures and decision tree complexity: a survey publication-title: Theor. Comput. Sci. doi: 10.1016/S0304-3975(01)00144-X – volume: 48 start-page: 778 issue: 4 year: 2001 end-page: 797 ident: CR4 article-title: Quantum lower bounds by polynomials publication-title: J. ACM doi: 10.1145/502090.502097 – ident: CR5 – ident: CR8 – ident: 9935_CR2 doi: 10.1145/2897518.2897644 – ident: 9935_CR5 doi: 10.1109/SFCS.1987.30 – volume: 8 start-page: 1 year: 2016 ident: 9935_CR11 publication-title: Chicago Journal Of Theoretical Computer Science doi: 10.4086/cjtcs.2016.008 – ident: 9935_CR8 – volume: 74 start-page: 313 issue: 3 year: 2008 ident: 9935_CR1 publication-title: J. Comput. Syst. Sci. doi: 10.1016/j.jcss.2007.06.020 – ident: 9935_CR10 doi: 10.1109/CCC.2010.31 – volume: 9 start-page: 385 year: 1990 ident: 9935_CR14 publication-title: Combinatorica doi: 10.1007/BF02125350 – ident: 9935_CR3 – volume: 48 start-page: 778 issue: 4 year: 2001 ident: 9935_CR4 publication-title: J. ACM doi: 10.1145/502090.502097 – ident: 9935_CR13 doi: 10.1145/2422436.2422485 – ident: 9935_CR12 doi: 10.1145/73007.73038 – volume: 36 start-page: 265 issue: 3 year: 2016 ident: 9935_CR7 publication-title: Combinatorica doi: 10.1007/s00493-014-3189-x – volume: 288 start-page: 21 issue: 1 year: 2002 ident: 9935_CR6 publication-title: Theor. Comput. Sci. doi: 10.1016/S0304-3975(01)00144-X – ident: 9935_CR9 doi: 10.1109/PSCT.1987.10319267 |
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| Snippet | In this work we investigate the problem of quadratically tightly approximating the randomized query complexity of Boolean functions
R
(
f
). The certificate... In this work we investigate the problem of quadratically tightly approximating the randomized query complexity of Boolean functions R(f). The certificate... |
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| StartPage | 101 |
| SubjectTerms | Boolean algebra Boolean functions Communication Complexity Computational Complexity Computer Science Computer Science Symposium in Russia Error analysis Queries Randomization Theory of Computation Upper bounds |
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| Title | Quadratically Tight Relations for Randomized Query Complexity |
| URI | https://link.springer.com/article/10.1007/s00224-019-09935-x https://www.proquest.com/docview/2270288018 https://hal.science/hal-02351409 |
| Volume | 64 |
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