Graph-theoretic arguments in low-level complexity

  • Leslie G. Valiant
Invited Lectures
Part of the Lecture Notes in Computer Science book series (LNCS, volume 53)


We have surveyed one approach to understanding complexity issues for certain easily computable natural functions. Shifting graphs have been seen to account accurately and in a unified way for the superlinear complexity of several problems for various restricted models of computation. To attack "unrestricted" models (in the present context combinational circuits or straight-line arithmetic programs,) a first attempt, through superconcentrators, fails to provide any lower bounds although it does give counter-examples to alternative approaches. The notion of rigidity, however, does offer for the first time a reduction of relevant computational questions to noncomputional properties. The "reduction" consists of the conjunction of Corollary 6.3 and Theorem 6.4 which show that "for most sets of linear forms over the reals the stated algebraic and combinatorial reasons account for the fact that they cannot be computed in linear time and depth O(log n) simultaneously." We have outlined some problem areas which our preliminary results raise, and feel that further progress on most of these is humanly feasible. We would be interested in alternative approaches also.

Problem 6 Propose reductions of relevant complexity issues to noncomputational properties, that are more promising or tractable than the ones above.


Linear Form Discrete Fourier Transform Turing Machine Input Node Combinational Circuit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aho, A.V., Hopcroft, J.E. and Ullman, J.D., The Design and Analysis of Computer Algorithms, Addison Wesley, 1974.Google Scholar
  2. 2.
    Beneš, V.E., Mathematical Theory of Connecting Networks and Telephone Traffic. Academic Press, New York, 1965.Google Scholar
  3. 3.
    Borodin, A.B. and Munro, I. The Complexity of Algebraic and Numeric Problems, American Elsevier, 1975.Google Scholar
  4. 4.
    Ehrenfeucht, A. Practical decidability. Report CU-CS-008-72, Univ. of Colorado (1972).Google Scholar
  5. 5.
    Erdös, P., Graham and Szemerédi, Ě. On sparse graphs with dense long paths. Comp. and Maths. with Appls., 1, (1975) 365–369.CrossRefGoogle Scholar
  6. 6.
    Fischer, M.J. and Rabin, M.O. Super-exponential complexity of Presburger arithmetic. MACTR43, Project MAC, MIT, (1974).Google Scholar
  7. 7.
    Hopcroft, J.E., Paul, W.J. and Valiant, L.G. Time versus space and related problems. Proc. 16th Symp. on Foundations of Computer Science, Berkeley, (1975) 57–64.Google Scholar
  8. 8.
    Hartmanis, J., Lewis, P.M. and Stearns, R.E. Classification of Computations by time and memory requirements. Proc. IFIP Congress 1965, Spartan, N.Y., 31–35.Google Scholar
  9. 9.
    Hyafil, L. and Kung, H.T. The complexity of parallel evaluation of linear recurrence. Proc. 7th ACM Symp. on Theory of Computing (1975) 12–22.Google Scholar
  10. 10.
    Margulis, G.A. Explicit constructions of Concentrators, Problemy Peredachi Informatsii, 9: 4(1973) 71–80.Google Scholar
  11. 11.
    Meyer, A.R. and Stockmeyer, L.J. The word problem for regular expressions with squaring requires exponential space. Proc. 13th IEEE Symp. on Switching and Automata Theory, (1972)125–129.Google Scholar
  12. 12.
    Paterson, M.S., Fischer, M.J. and Meyer A.R. An improved overlap argument for on-line multiplication. SIAM-AMS Proceedings Vol 7, (1974) 97–111Google Scholar
  13. 13.
    Paterson, M.S. and Valiant, L.G. Circuit size is nonlinear in depth. Theoretical Computer Science 2 (1976) 397–400.CrossRefGoogle Scholar
  14. 14.
    Paul, W.J. A 2.5N Lower bound for the combinational complexity of boolean functions. Proc. 7th ACM Symp. on Theory of Computing, (1975) 27–36.Google Scholar
  15. 15.
    Paul, W.J., Tarjan, R.E. and Celoni, J.R. Space bounds for a game on graphs. Proc. 8th ACM Symp. on Theory of Computing, (1976) 149–160.Google Scholar
  16. 16.
    Pippenger, N. The complexity theory of switching networks. Ph.D. Thesis, Dept. of Elect. Eng., MIT, (1973).Google Scholar
  17. 17.
    Pippenger, N. Superconcentrators. RC5937. IBM Yorktown Heights (1976).Google Scholar
  18. 18.
    Pippenger, N. and Valiant, L.G. Shifting graphs and their applications. JACM 23 (1976) 423–432.CrossRefGoogle Scholar
  19. 19.
    Schnorr, C.P. Zwei lineare Schranken fur die Komplexität Boolischer Funktionen, Computing, 13 (1974) 155–171.Google Scholar
  20. 20.
    Stockmeyer, L.J. and Meyer, A.R. Inherent computational complexity of decision problems in logic and automata theory. Lecture Notes in Computer Science (to appear), SpringerGoogle Scholar
  21. 21.
    Strassen, V. Die Berechnungkomplexität von elementar symmetrichen Funktionen und von Interpolationskoeffizienten. Numer. Math 20 (1973) 238–251.CrossRefGoogle Scholar
  22. 22.
    Strassen, V. Vermeidung von Divisionen, J.Reine Angew.Math., 264, (1973), 184–202.Google Scholar
  23. 23.
    Valiant, L.G. On non-linear lower bounds in computational complexity. Proc. 7th ACM Symp. on Theory of Computing, (1975) 45–53.Google Scholar
  24. 24.
    Valiant, L.G. Universal circuits. Proc. 8th ACM Symp. on Theory of Computing, (1976) 196–203.Google Scholar
  25. 25.
    Valiant, L.G. Some conjectures relating to superlinear lower bounds. TR85, Dept. of Comp. Sci., Univ. of Leeds (1976).Google Scholar
  26. 26.
    Winograd, S. On the number of multiplications necessary to compute certain functions. Comm. on Pure and App. Math. 23 (1970) 165–179.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1977

Authors and Affiliations

  • Leslie G. Valiant
    • 1
  1. 1.Computer Science DepartmentUniversity of EdinburghEdinburghScotland

Personalised recommendations