Article

Two applications of information complexity

Authors:

Ravi Kumar, and

D. SivakumarAuthors Info & Claims

STOC '03: Proceedings of the thirty-fifth annual ACM symposium on Theory of computing

June 2003

Pages 673 - 682

https://doi.org/10.1145/780542.780640

Published: 09 June 2003 Publication History

Abstract

We show the following new lower bounds in two concrete complexity models:

(1) In the two-party communication complexity model, we show that the tribes function on n inputs[6] has two-sided error randomized complexity Ω(n), while its nondeterminstic complexity and co-nondeterministic complexity are both Θ(√n). This separation between randomized and nondeterministic complexity is the best possible and it settles an open problem in Kushilevitz and Nisan[17], which was also posed by Beame and Lawry[5].

(2) In the Boolean decision tree model, we show that the recursive majority-of-three function on 3^h inputs has randomized complexity Ω((7/3)^h). The deterministic complexity of this function is Θ(3^h), and the nondeterministic complexity is Θ(2^h). Our lower bound on the randomized complexity is a substantial improvement over any lower bound for this problem that can be obtained via the techniques of Saks and Wigderson [23], Heiman and Wigderson[14], and Heiman, Newman, and Wigderson[13]. Recursive majority is an important function for which a class of natural algorithms known as directional algorithms does not achieve the best randomized decision tree upper bound.

These lower bounds are obtained using generalizations of information complexity, which quantifies the minimum amount of information that will have to be revealed about the inputs by every correct algorithm in a given model of computation.

References

[1]

F. Ablayev. Lower bounds for one-way probabilistic communication complexity and their application to space complexity. Theoretical Computer Science, 157(2):139--159, 1996.

Digital Library

[2]

A. V. Aho, J. D. Ullman, and M. Yannakakis. On notions of information transfer in VLSI circuits. In Proc. 15th Annual ACM Symposium on the Theory of Computing, pages 133--139, 1983.

Digital Library

[3]

R. Bar-Yehuda, B. Chor, E. Kushilevitz, and A. Orlitsky. Privacy, additional information, and communication. IEEE Transactions on Information Theory, 39(6):1930--1943, 1993.

Digital Library

[4]

Z. Bar-Yossef, T. Jayram, R. Kumar, and D. Sivakumar. An information statistics approach to data stream and communication complexity. In Proc. 43rd Annual IEEE Symposium on Foundations of Computer Science, pages 209--218, 2002.

Digital Library

[5]

P. Beame and J. Lawry. Randomized versus nondeterministic communication complexity. In Proc. 24th Annual ACM Symposium on the Theory of Computing, pages 188--199, 1992.

Digital Library

[6]

M. Ben-Or and N. Linial. Collective coin flipping. In S. Micali, editor, Randomness and Computation, pages 91--115. JAI Press, 1990.

[7]

M. Blum and R. Impagliazzo. General oracle and oracle classes. In Proc. 28th Annual IEEE Symposium on Foundations of Computer Science, pages 118--126, 1987.

Digital Library

[8]

H. Buhrman and R. de~Wolf. Complexity measures and decision tree complexity: A survey. Theoretical Computer Science, 288(1):21--43, 2002.

Digital Library

[9]

A. Chakrabarti, Y. Shi, A. Wirth, and A. C.-C. Yao. Informational complexity and the direct sum problem for simultaneous message complexity. In Proc. 42nd IEEE Annual Symposium on Foundations of Computer Science, pages 270--278, 2001.

Digital Library

[10]

T. M. Cover and J. A. Thomas. Elements of Information Theory. John Wiley & Sons, Inc., 1991.

Digital Library

[11]

M. Furer. The power of randomness for communication complexity. In Proc. 19th Annual ACM Symposium on the Theory of Computing, pages 178--181, 1987.

Digital Library

[12]

J. Hartmanis and L. Hemachandra. Complexity classes without machines: On complete languages for UP. Theoretical Computer Science, 58:129--142, 1988.

Digital Library

[13]

R. Heiman, I. Newman, and A. Wigderson. On read-once threshold formulae and their randomized decision tree complexity. Theoretical Computer Science, 107(1):63--76, 1993.

Digital Library

[14]

R. Heiman and A. Wigderson. Randomized vs. deterministic decision tree complexity for read-once Boolean functions. Computational Complexity, 1:311--329, 1991.

[15]

J. Hromkovic and G. Schnitger. Nondeterministic communication with a limited number of advice bits. In Proc. 28th Annual ACM Symposium on the Theory of Computing, pages 551--560, 1996.

Digital Library

[16]

M. Karchmer and A. Wigderson. Monotone circuits for connectivity require super-logarithmic depth. SIAM Journal on Discrete Mathematics, 3(2):255--265, 1990.

[17]

E. Kushilevitz and N. Nisan. Communication Complexity. Cambridge University Press, 1997.

Digital Library

[18]

J. Lin. Divergence measures based on the Shannon entropy. IEEE Transactions on Information Theory, 37(1):145--151, 1991.

Digital Library

[19]

I. Newman. Private vs. common random bits in communication complexity. Information Processing Letters, 39:67--71, 1991.

Digital Library

[20]

N. Nisan. CREW PRAMs and decision trees. SIAM Journal on Computing, 20(6):999--1007, 1991.

Digital Library

[21]

R. Raz and A. Wigderson. Monotone circuits for matching require linear depth. Journal of the ACM, 39(3):736--744, 1992.

Digital Library

[22]

M. Saks and X. Sun. Space lower bounds for distance approximation in the data stream model. In Proc. of the 34th Annual ACM Symposium on Theory of Computing, pages 360--369, 2002.

Digital Library

[23]

M. Saks and A. Wigderson. Probabilistic Boolean decision trees and the complexity of evaluating game trees. In Proc. 27th IEEE Symposium on Foundations of Computer Science, pages 29--38, 1986.

Digital Library

[24]

M. Santha. On the Monte Carlo Boolean decision tree complexity of read-once formulae. Random Structures and Algorithms, 6(1):75--88, 1995.

[25]

M. Snir. Lower bounds for probabilistic linear decision trees. Theoretical Computer Science, 38:69--82, 1985.

[26]

G. Tardos. Query complexity, or why is it difficult to separate NPA ∩ co-NPA from PA by a random oracle. Combinatorica, 9:385--392, 1990.

[27]

A. C.-C. Yao. Some complexity questions related to distributive computing. In Proc. 11th Annual ACM Symposium on Theory of Computing, pages 209--213, 1979.

Digital Library

[28]

A. C.-C. Yao. The entropic limitations on VLSI computations (extended abstract). In Proc. 13th Annual ACM Symposium on Theory of computing, pages 308--311, 1981.

Digital Library

Cited By

Li Y(2022)Trading information complexity for error II: The case of a large error and the external information complexityInformation and Computation10.1016/j.ic.2022.104952(104952)Online publication date: Aug-2022
https://doi.org/10.1016/j.ic.2022.104952
Watson T(2020)Communication Complexity with Small Advantagecomputational complexity10.1007/s00037-020-00192-w29:1Online publication date: 20-Apr-2020
https://doi.org/10.1007/s00037-020-00192-w
Assadi SChen YKhanna SCharikar MCohen E(2019)Polynomial pass lower bounds for graph streaming algorithmsProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing10.1145/3313276.3316361(265-276)Online publication date: 23-Jun-2019
https://dl.acm.org/doi/10.1145/3313276.3316361
Show More Cited By

Index Terms

Two applications of information complexity
1. Theory of computation
  1. Computational complexity and cryptography
    1. Complexity classes

Recommendations

An information complexity approach to extended formulations
STOC '13: Proceedings of the forty-fifth annual ACM symposium on Theory of Computing

We prove an unconditional lower bound that any linear program that achieves an O(n^1-ε) approximation for clique has size 2^{Ω(n^ε)}. There has been considerable recent interest in proving unconditional lower bounds against any linear program. Fiorini et al. ...
Read More
Symmetry of information from meta-complexity
CCC '22: Proceedings of the 37th Computational Complexity Conference

Symmetry of information for time-bounded Kolmogorov complexity is a hypothetical inequality that relates time-bounded Kolmogorov complexity and its conditional analogue. In 1992, Longpré and Watanabe showed that symmetry of information holds if NP is ...
Read More
The Round Complexity of Two-Party Random Selection

We study the round complexity of two-party protocols for generating a random $n$-bit string such that the output is guaranteed to have bounded “bias,” even if one of the two parties deviates from the protocol (possibly using unlimited computational ...
Read More

Comments

Information & Contributors

Information

Published In

cover image ACM Conferences

STOC '03: Proceedings of the thirty-fifth annual ACM symposium on Theory of computing

June 2003

740 pages

ISBN:1581136749

DOI:10.1145/780542

Conference Chair:
Lawrence L. Larmore
University of Nevada Las Vegas, Las Vegas, NV
,
Program Chair:
Michel X. Goemans
Massachusetts Institute of Technology, Cambridge, MA

Copyright © 2003 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Sponsors

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 09 June 2003

Permissions

Request permissions for this article.

Request Permissions

Check for updates

Qualifiers

Article

Conference

STOC03

Sponsor:

STOC03: The 35th Annual ACM Symposium on Theory of Computing

June 9 - 11, 2003

CA, San Diego, USA

Acceptance Rates

STOC '03 Paper Acceptance Rate 80 of 270 submissions, 30%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

62
Total Citations
View Citations
1,111
Total Downloads

Downloads (Last 12 months)45
Downloads (Last 6 weeks)10

Other Metrics

View Author Metrics

Citations

Cited By

Li Y(2022)Trading information complexity for error II: The case of a large error and the external information complexityInformation and Computation10.1016/j.ic.2022.104952(104952)Online publication date: Aug-2022
https://doi.org/10.1016/j.ic.2022.104952
Watson T(2020)Communication Complexity with Small Advantagecomputational complexity10.1007/s00037-020-00192-w29:1Online publication date: 20-Apr-2020
https://doi.org/10.1007/s00037-020-00192-w
Assadi SChen YKhanna SCharikar MCohen E(2019)Polynomial pass lower bounds for graph streaming algorithmsProceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing10.1145/3313276.3316361(265-276)Online publication date: 23-Jun-2019
https://dl.acm.org/doi/10.1145/3313276.3316361
Langberg MLi SMani Jayaraman SRudra ASuciu DSkritek SKoch C(2019)Topology Dependent Bounds For FAQsProceedings of the 38th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems10.1145/3294052.3319686(432-449)Online publication date: 25-Jun-2019
https://dl.acm.org/doi/10.1145/3294052.3319686
Watson TServedio R(2018)Communication complexity with small advantageProceedings of the 33rd Computational Complexity Conference10.5555/3235586.3235595(1-17)Online publication date: 22-Jun-2018
https://dl.acm.org/doi/10.5555/3235586.3235595
Kannan SMossel ESanyal SYaroslavtsev GServedio R(2018)Linear sketching over F2Proceedings of the 33rd Computational Complexity Conference10.5555/3235586.3235594(1-37)Online publication date: 22-Jun-2018
https://dl.acm.org/doi/10.5555/3235586.3235594
Laplante SLaurière MNolin ARoland JSenno G(2018)Robust Bell inequalities from communication complexityQuantum10.22331/q-2018-06-07-722(72)Online publication date: 7-Jun-2018
https://doi.org/10.22331/q-2018-06-07-72
Göös MPitassi TWatson T(2018)The Landscape of Communication Complexity ClassesComputational Complexity10.1007/s00037-018-0166-627:2(245-304)Online publication date: 1-Jun-2018
https://dl.acm.org/doi/10.1007/s00037-018-0166-6
Göös MJayram T(2016)A composition theorem for conical juntasProceedings of the 31st Conference on Computational Complexity10.5555/2982445.2982450(1-16)Online publication date: 29-May-2016
https://dl.acm.org/doi/10.5555/2982445.2982450
Göös MPitassi TWatson T(2016)Zero-Information Protocols and Unambiguity in Arthur---Merlin CommunicationAlgorithmica10.1007/s00453-015-0104-976:3(684-719)Online publication date: 1-Nov-2016
https://dl.acm.org/doi/10.1007/s00453-015-0104-9
Show More Cited By

View Options

Get Access

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

View options

PDF

View or Download as a PDF file.

eReader

View online with eReader.

Media

Figures

Other

Tables

View Table of Contents