The complexity of graph connectivity

  • Avi Wigderson
Invited Lectures
Part of the Lecture Notes in Computer Science book series (LNCS, volume 629)


In this paper we survey the major developments in understanding the complexity of the graph connectivity problem in several computational models, and highlight some challenging open problems.


Boolean Function Turing Machine Input Graph Universal Sequence Circuit Class 
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. [A1]
    M. Ajtai, On the complexity of the pigeonhole principle, Proc. of the 29th FOCS, pp. 346–355, 1988.Google Scholar
  2. [A2]
    M. Ajtai, First-order definability on finite structures, Annals of Pure and Applied Logic, 45, pp. 211–225, 1989.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [AB]
    M. Ajtai and M. Ben-Or, A theorem on probabilistic constant-depth computation, Proc. of the 16th STOC, pp. 471–474, 1984.Google Scholar
  4. [AF]
    M. Ajtai and R. Fagin, Reachability is harder for directed than for undirected finite graphs, The journal of Symbolic Logic, Vol 55, No 1, pp. 113–150, 1990.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [AKS]
    M. Ajtai, J. Komlos, E. Szemeredi, Deterministic simulation in logspace, Proc. of the 19th STOC, pp. 132–140, 1987.Google Scholar
  6. [AK+]
    R. Aleliunas, R. M. Karp, R. J. Lipton, L. Lovasz, C. Rackoff, Random walks, universal traversal sequences, and the complexity of maze problems, Proc. of the 20th FOCS, pp. 218–223, 1979.Google Scholar
  7. [BeSi]
    P. Berman and J. Simon, Lower bounds for graph threading by probabilistic machines, Proc. of the 24th FOCS, pp. 304–311, 1983.Google Scholar
  8. [Bo]
    B. Bollobas, Extremal Graph Theory, Academic Press, 1978.Google Scholar
  9. [BBRS]
    G. Barnes, J. F. Buss, W. L. Ruzzo and B. Schieber, A sublinear space, polynomial time algorithm for directed s-t connectivity, Technical report 92-03-05, Dept. of Computer Science, University of Washington, 1992.Google Scholar
  10. [BC+]
    A. Borodin, S. A. Cook, P. W. Dymond, W. L. Ruzzo and M. Tompa, Two applications of inductive counting for complementation problems, SIAM J. on Computing, Vol 18, pp. 559–578, 1989.zbMATHMathSciNetCrossRefGoogle Scholar
  11. [BI+]
    P. Beame, R. Impagliazzo, J. Krajicek, T. Pitassi, P. Pudlak and A. Woods, Exponential lower bounds for the pigeonhole principle, Proc. of the 24th STOC, pp. 200–220.Google Scholar
  12. [BM]
    M. Blum and S. Micali, How to generate cryptographically strong sequences of pseudo-random bits, SIAM J. on Computing, 13, 4, pp. 850–864, 1984.MathSciNetCrossRefGoogle Scholar
  13. [BNS]
    L. Babai, N. Nisan and M. Szegedy, Multi-party protocols and logspacehard pseudo-random sequences, Proc. of the 21st STOC, pp.1–11, 1989.Google Scholar
  14. [BR]
    G. Barnes and W. L. Ruzzo, Deterministic algorithms for undirected st-connectivity using polynomial time and sublinear space, Proc. 23rd STOC, pp. 43–53, 1991.Google Scholar
  15. [BS]
    R. Boppana and M. Sipser, The complexity of finite functions, Handbook of Theoretical Compluter Science, Vol. A, van Leeuwen (ed.), MIT Press/Elsvier, pp. 759–804, 1990.Google Scholar
  16. [CaWe]
    L. Carter and M. Wegman, Universal hash functions, J. of Computer Systems and Sciences, 18, 2, pp. 143–154, 1979.zbMATHMathSciNetCrossRefGoogle Scholar
  17. [Co]
    S. A. Cook, A taxonomy of problems with fast parallel algorithms, Information and Computation, 64, pp. 2–22, 1985.zbMATHGoogle Scholar
  18. [CW]
    A. Cohen and A. Wigderson, Dispersers, deterministic amplification and weak random sources, Proc. of the 30th FOCS, pp. 14–19, 1989.Google Scholar
  19. [CoWi]
    D. Coppersmith and S. Winograd, Matrix multiplication via arithmetic progressions, Proc. of the 19th STOC, pp. 1–6, 1987.Google Scholar
  20. [CFL]
    A. Chandra, M. Furst and R. J. Lipton, Multi-party protocols, Proc. of the 15th STOC, pp. 94–99, 1983.Google Scholar
  21. [CKR]
    M. Chrobak, H. Karloff, T. Radzik, Connectivity vs. reachability, Information and Computation, Vol 91, No 2, pp. 177–188, 1991.zbMATHMathSciNetCrossRefGoogle Scholar
  22. [CM]
    S. Cook and P. McKenzie, manuscript, 1986.Google Scholar
  23. [cr]1992
    Springer-VerlagS. A. Cook and C. W. Rackoff, Space lower bounds for maze threadability on restricted machines, SIAM J. on Computing, Vol 9, No 3, pp. 636–652, 1980.zbMATHMathSciNetCrossRefGoogle Scholar
  24. [E]
    H. B. Enderton, A Mathematical Introduction to Logic, Academic Press, 1972.Google Scholar
  25. [Eh]
    A. Ehrenfeucht, An application of games to the completeness problem for formalized theories, Fund. Math., 49, pp. 129–141, 1961.zbMATHMathSciNetGoogle Scholar
  26. [Fa]
    R. Fagin, Monadic generalized spectra, Seitschrift fur Mathematische Logik und Grundlagen der Mathematik, Vol 21, pp. 89–96, 1975.zbMATHMathSciNetGoogle Scholar
  27. [Fr]
    R. Fraisse, Sur les classifications des systems de relations, Publications Scientifiques de l'Université d'Alger, Vol 1. pp. 35–182, 1954.MathSciNetGoogle Scholar
  28. [FSS]
    M. Furst, J. Saxe and M. Sipser, Parity, circuits and the polynomial-time hierarchy, Math System Theory 17, pp. 13–27, 1984.zbMATHMathSciNetCrossRefGoogle Scholar
  29. [GS1]
    M. Grigni and M. Sipser, Monotone complexity, Proceedings of LMS workshop on Boolean function complexity, Durham, M. Paterson (Ed.), Cambridge University Press, 1990.Google Scholar
  30. [GS2]
    M. Grigni and M. Sipser, Monotone separation of Logspace from NC 1, Proc. of the 6th Structures in Complexity Theory conference, pp. 294–298, 1991.Google Scholar
  31. [H]
    J. Hastad, Computational limitations of small-depth circuits, The MIT Press, 1987.Google Scholar
  32. [Is]
    S. Istrail, Polynomial traversing sequences for cycles are constructible, Proc. of the 20th STOC, pp. 491–503, 1988.Google Scholar
  33. [I1]
    N. Immerman, Descriptive and computational complexity, Computational Complexity Theory, J. Hartmanis (Ed.), Proc. Symp. Applied Math. 38, American Mathematical Society, pp. 75–91, 1989.Google Scholar
  34. [I2]
    N. Immerman, Nondeterministic space is closed under complementation, SIAM J. on Computing, 17, pp. 935–938, 1988.zbMATHMathSciNetCrossRefGoogle Scholar
  35. [I3]
    R. Implagliazzo and D. Zuckerman, How to recycle random bits, Proc. of the 30th FOCS, pp. 248–253, 1989.Google Scholar
  36. [J]
    D. Johnson, A catalog of complexity classes, Handbook of Theoretical Compluter Science, Vol. A, van Leeuwen (ed.), MIT Press/ Elsvier, pp. 67–162, 1990.Google Scholar
  37. [Ka]
    P. Kanellakis, Private communication, 1986.Google Scholar
  38. [KSS]
    J. Kahn, M. Saks, D. Sturtevant, A topological approach to evasiveness, Combinatorica 4, pp. 297–306, 1984.zbMATHMathSciNetGoogle Scholar
  39. [KW]
    M. Karchmer and A. Wigderson, Monotone circuits for connectivity require super-logarithmic depth, SIAM J. on Discrete Mathematics, Vol 3, No 2. pp. 255–265, 1990.zbMATHMathSciNetCrossRefGoogle Scholar
  40. [LP]
    H. Lewis and C. Papadimitriu, Symmetric space-bounded computation, Theoretical Computer Science 25, pp. 130–143, 1982.Google Scholar
  41. [N1]
    N. Nisan, Pseudo-random generators for space-bounded computation, Proc. of the 22nd STOC, pp. 204–212, 1990.Google Scholar
  42. [N2]
    N. Nisan, RL ∈ SC, Proc. of the 24th STOC, pp. 619–623, 1992.Google Scholar
  43. [NN]
    J. Naor and M. Naor, Small-bias probability spaces: efficient constuctions and applications, Proc. of the 22nd STOC, pp. 213–223, 1990.Google Scholar
  44. [NSW]
    N. Nisan, E. Szemeredi and A. Wigderson, Undirected connectivity in O(log1.5 n) space, submitted to FOCS '92.Google Scholar
  45. [NW]
    N. Nisan and A. Wigderson, Hardness vs. Randomness, Proc. of the 29th FOCS, pp. 2–12, 1988.Google Scholar
  46. [RW1]
    R. Raz and A. Wigderson, Probabilistic communication complexity of Boolean relations, Proc. of the 30th FOCS, pp. 562–567, 1989.Google Scholar
  47. [RW2]
    R. Raz and A. Wigderson, Monotone circuits for matching require linear depth, Proc. of the 22nd STOC, pp. 287–292, 1990.Google Scholar
  48. [Re]
    J. H. Reif, Symmetric complementation, Proc. of the 14th STOC, pp. 201–214, 1982.Google Scholar
  49. [S]
    R. Szelepcsenyi, The method of forcing for nondeterministic automata, Bull. of the European Ass. of Theoretical Computer Science, 33, pp. 96–100, 1987.zbMATHGoogle Scholar
  50. [Sa]
    W. Savitch, Relashionships between nondeterministic and deterministic tape complexities, Journal of Computer Systems and Sciences, 4, pp. 177–192, 1970.zbMATHMathSciNetGoogle Scholar
  51. [SV]
    S. Skyum and L. Valiant, A complexity theory based on Boolean algebra, Proc. of the 22nd FOCS, pp. 244–253, 1981.Google Scholar
  52. [T]
    M. Tompa, Two familiar transitive closure algorithms which admit no polynomial time, sublinear space implementations, SIAM J. on Computing, 11, 1, pp. 130–137, 1982.zbMATHMathSciNetCrossRefGoogle Scholar
  53. [Y1]
    A. C. Yao, Some complexity questions related to distributive computing, Proc. of the 11th STOC, pp. 209–213, 1979.Google Scholar
  54. [Y2]
    A. C. Yao, Private communication.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Avi Wigderson
    • 1
  1. 1.Hebrew University and Princeton UniversityUSA

Personalised recommendations