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On non-linear lower bounds in computational complexity

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Leslie G. ValiantAuthors Info & Claims

STOC '75: Proceedings of the seventh annual ACM symposium on Theory of computing

Pages 45 - 53

https://doi.org/10.1145/800116.803752

Published: 05 May 1975 Publication History

Abstract

The purpose of this paper is to explore the possibility that purely graph-theoretic reasons may account for the superlinear complexity of wide classes of computational problems. The results are therefore of two kinds: reductions to graph theoretic conjectures on the one hand, and graph theoretic results on the other.

We show that the graph of any algorithm for any one of a number of arithmetic problems (e.g. polynomial multiplication, discrete Fourier transforms, matrix multiplication) must have properties closely related to concentration networks.

References

[1]

Aho, A.V., Hirschberg, D.S. and Ullman, J.D. Bounds on the Complexity of the longest subsequence problem. Proc. 15th Symp. on SWAT 104-109, (1974).

Digital Library

[2]

Aho, A.V., Hopcroft, J.E. and Ullman, J.D. The Design and Analysis of Computer Algorithms. Addison-Wesley, Reading, Mass. (1974).

Digital Library

[3]

Borodin, A. and Munro, I. Computational Complexity of Algebraic and Numeric Problems. American Elsevier. (1975).

[4]

Cook, S.A. and Reckhow, R. A. Time bounded random access machines. JCSS 7, 354-375, (1973).

Digital Library

[5]

Fischer, M.J. and Paterson, M.S. String-matching and other products. MAC Technical memorandum 41, MIT, (1974).

Digital Library

[6]

Fischer, M.J. and Pippinger, N. To appear.

[7]

Floyd, R.W. Permuting information in idealized two-level storage. In Complexity of Computer Computations, R. E. Miller and J. W. Thatcher (eds.), Plenum Press (1972).

[8]

Knuth, D.E. The Art of Computer Programming, Vol. 3, Addison Wesley, Reading, Mass. (1973).

Digital Library

[9]

Morris, J.H. and Pratt, V.R. A linear pattern-matching algorithm. TR-40, Computer Center, University of California at Berkeley (1970).

[10]

Pinsker, M.S. On the complexity of a concentrator. 7th International Teletraffic Congress, Stockholm. (1973).

[11]

Pippenger, N. The complexity theory of switching networks. Technical Report 487, Res. Lab. of Electronics, MIT, (1973).

[12]

Savage, J.E. The Complexity of Computing, Chapter 2, (manuscript).

Digital Library

[13]

Strassen, V. Die Berechnungkomplexitat von elementarsymmetrischen Funktionen und von Interpolationskoeffizienten. Numer. Math 20, (1973).

[14]

Valiant, L.G. Parallelism in comparison problems. SIAM J. on Computing, (to appear).

[15]

Wagner, R. A., and Fischer, M. J. The string to string correction problem. JACM 21:1, 168-173, (1974).

Digital Library

[16]

Waksman, A. A permutation network. JACM 15, 159-163 (1968).

Digital Library

[17]

Yao, F.F. personal communication.

[18]

Yao, A.C.-C., and Yao, F.F. Lower bounds on merging networks. Manuscript (1974).

Cited By

Drucker ALi Y(2023)On the Minimum Depth of Circuits with Linear Number of Wires Encoding Good CodesComputing and Combinatorics10.1007/978-3-031-49193-1_30(392-403)Online publication date: 9-Dec-2023
https://doi.org/10.1007/978-3-031-49193-1_30
Volk BKumar M(2021)Lower Bounds for Matrix Factorizationcomputational complexity10.1007/s00037-021-00205-230:1Online publication date: 2-Apr-2021
https://doi.org/10.1007/s00037-021-00205-2
Dutta P(2021)Real $$\tau $$-Conjecture for Sum-of-Squares: A Unified Approach to Lower Bound and DerandomizationComputer Science – Theory and Applications10.1007/978-3-030-79416-3_5(78-101)Online publication date: 17-Jun-2021
https://doi.org/10.1007/978-3-030-79416-3_5
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Index Terms

On non-linear lower bounds in computational complexity
1. Theory of computation
  1. Design and analysis of algorithms
  2. Models of computation
    1. Computability
      1. Turing machines

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cover image ACM Conferences

STOC '75: Proceedings of the seventh annual ACM symposium on Theory of computing

May 1975

265 pages

ISBN:9781450374194

DOI:10.1145/800116

Copyright © 1975 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]

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Published: 05 May 1975

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STOC '75 Paper Acceptance Rate 31 of 87 submissions, 36%;

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

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Cited By

Drucker ALi Y(2023)On the Minimum Depth of Circuits with Linear Number of Wires Encoding Good CodesComputing and Combinatorics10.1007/978-3-031-49193-1_30(392-403)Online publication date: 9-Dec-2023
https://doi.org/10.1007/978-3-031-49193-1_30
Volk BKumar M(2021)Lower Bounds for Matrix Factorizationcomputational complexity10.1007/s00037-021-00205-230:1Online publication date: 2-Apr-2021
https://doi.org/10.1007/s00037-021-00205-2
Dutta P(2021)Real $$\tau $$-Conjecture for Sum-of-Squares: A Unified Approach to Lower Bound and DerandomizationComputer Science – Theory and Applications10.1007/978-3-030-79416-3_5(78-101)Online publication date: 17-Jun-2021
https://doi.org/10.1007/978-3-030-79416-3_5
Kumar MVolk BAceto L(2020)Lower bounds for matrix factorizationProceedings of the 35th Computational Complexity Conference10.4230/LIPIcs.CCC.2020.5(1-20)Online publication date: 28-Jul-2020
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2020.5
Cecchetti EFisch BMiers IJuels ACavallaro LKinder JWang XKatz J(2019)PIEsProceedings of the 2019 ACM SIGSAC Conference on Computer and Communications Security10.1145/3319535.3354231(1351-1367)Online publication date: 6-Nov-2019
https://dl.acm.org/doi/10.1145/3319535.3354231
Gal AHansen KKoucky MPudlak PViola E(2013)Tight Bounds on Computing Error-Correcting Codes by Bounded-Depth Circuits With Arbitrary GatesIEEE Transactions on Information Theory10.1109/TIT.2013.227027559:10(6611-6627)Online publication date: 1-Oct-2013
https://dl.acm.org/doi/10.1109/TIT.2013.2270275
Chung F(2013)On Concentrators, Superconcentrators, Generalizers, and Nonblocking NetworksBell System Technical Journal10.1002/j.1538-7305.1979.tb02972.x58:8(1765-1777)Online publication date: 29-Jul-2013
https://doi.org/10.1002/j.1538-7305.1979.tb02972.x
Gál AHansen KKoucký MPudlák PViola EKarloff HPitassi T(2012)Tight bounds on computing error-correcting codes by bounded-depth circuits with arbitrary gatesProceedings of the forty-fourth annual ACM symposium on Theory of computing10.1145/2213977.2214023(479-494)Online publication date: 19-May-2012
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Wang YNgo HNguyen T(2012)Constructions of given-depth and optimal multirate rearrangeably nonblocking distributorsJournal of Combinatorial Optimization10.1007/s10878-011-9402-624:4(468-484)Online publication date: 1-Nov-2012
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Cheung HLau LLeung K(2011)Graph Connectivities, Network Coding, and Expander GraphsProceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science10.1109/FOCS.2011.55(190-199)Online publication date: 22-Oct-2011
https://dl.acm.org/doi/10.1109/FOCS.2011.55
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