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International Symposium on Mathematical Foundations of Computer Science

MFCS 1992: Mathematical Foundations of Computer Science 1992 pp 112–132Cite as

The complexity of graph connectivity

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 629))

Abstract

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.

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References

  1. M. Ajtai, On the complexity of the pigeonhole principle, Proc. of the 29th FOCS, pp. 346–355, 1988.

    Google Scholar 

  2. M. Ajtai, First-order definability on finite structures, Annals of Pure and Applied Logic, 45, pp. 211–225, 1989.

    Article  MATH  MathSciNet  Google Scholar 

  3. 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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  5. M. Ajtai, J. Komlos, E. Szemeredi, Deterministic simulation in logspace, Proc. of the 19th STOC, pp. 132–140, 1987.

    Google Scholar 

  6. 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. 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. B. Bollobas, Extremal Graph Theory, Academic Press, 1978.

    Google Scholar 

  9. 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. 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.

    Article  MATH  MathSciNet  Google Scholar 

  11. 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. 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.

    Article  MathSciNet  Google Scholar 

  13. 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. 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. 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. L. Carter and M. Wegman, Universal hash functions, J. of Computer Systems and Sciences, 18, 2, pp. 143–154, 1979.

    Article  MATH  MathSciNet  Google Scholar 

  17. S. A. Cook, A taxonomy of problems with fast parallel algorithms, Information and Computation, 64, pp. 2–22, 1985.

    MATH  Google Scholar 

  18. A. Cohen and A. Wigderson, Dispersers, deterministic amplification and weak random sources, Proc. of the 30th FOCS, pp. 14–19, 1989.

    Google Scholar 

  19. D. Coppersmith and S. Winograd, Matrix multiplication via arithmetic progressions, Proc. of the 19th STOC, pp. 1–6, 1987.

    Google Scholar 

  20. A. Chandra, M. Furst and R. J. Lipton, Multi-party protocols, Proc. of the 15th STOC, pp. 94–99, 1983.

    Google Scholar 

  21. M. Chrobak, H. Karloff, T. Radzik, Connectivity vs. reachability, Information and Computation, Vol 91, No 2, pp. 177–188, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  22. S. Cook and P. McKenzie, manuscript, 1986.

    Google Scholar 

  23. 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.

    Article  MATH  MathSciNet  Google Scholar 

  24. H. B. Enderton, A Mathematical Introduction to Logic, Academic Press, 1972.

    Google Scholar 

  25. A. Ehrenfeucht, An application of games to the completeness problem for formalized theories, Fund. Math., 49, pp. 129–141, 1961.

    MATH  MathSciNet  Google Scholar 

  26. R. Fagin, Monadic generalized spectra, Seitschrift fur Mathematische Logik und Grundlagen der Mathematik, Vol 21, pp. 89–96, 1975.

    MATH  MathSciNet  Google Scholar 

  27. R. Fraisse, Sur les classifications des systems de relations, Publications Scientifiques de l'Université d'Alger, Vol 1. pp. 35–182, 1954.

    MathSciNet  Google Scholar 

  28. M. Furst, J. Saxe and M. Sipser, Parity, circuits and the polynomial-time hierarchy, Math System Theory 17, pp. 13–27, 1984.

    Article  MATH  MathSciNet  Google Scholar 

  29. 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. 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. J. Hastad, Computational limitations of small-depth circuits, The MIT Press, 1987.

    Google Scholar 

  32. S. Istrail, Polynomial traversing sequences for cycles are constructible, Proc. of the 20th STOC, pp. 491–503, 1988.

    Google Scholar 

  33. 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. N. Immerman, Nondeterministic space is closed under complementation, SIAM J. on Computing, 17, pp. 935–938, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  35. R. Implagliazzo and D. Zuckerman, How to recycle random bits, Proc. of the 30th FOCS, pp. 248–253, 1989.

    Google Scholar 

  36. 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. P. Kanellakis, Private communication, 1986.

    Google Scholar 

  38. J. Kahn, M. Saks, D. Sturtevant, A topological approach to evasiveness, Combinatorica 4, pp. 297–306, 1984.

    MATH  MathSciNet  Google Scholar 

  39. 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.

    Article  MATH  MathSciNet  Google Scholar 

  40. H. Lewis and C. Papadimitriu, Symmetric space-bounded computation, Theoretical Computer Science 25, pp. 130–143, 1982.

    Google Scholar 

  41. N. Nisan, Pseudo-random generators for space-bounded computation, Proc. of the 22nd STOC, pp. 204–212, 1990.

    Google Scholar 

  42. N. Nisan, RL ∈ SC, Proc. of the 24th STOC, pp. 619–623, 1992.

    Google Scholar 

  43. 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. N. Nisan, E. Szemeredi and A. Wigderson, Undirected connectivity in O(log1.5 n) space, submitted to FOCS '92.

    Google Scholar 

  45. N. Nisan and A. Wigderson, Hardness vs. Randomness, Proc. of the 29th FOCS, pp. 2–12, 1988.

    Google Scholar 

  46. R. Raz and A. Wigderson, Probabilistic communication complexity of Boolean relations, Proc. of the 30th FOCS, pp. 562–567, 1989.

    Google Scholar 

  47. R. Raz and A. Wigderson, Monotone circuits for matching require linear depth, Proc. of the 22nd STOC, pp. 287–292, 1990.

    Google Scholar 

  48. J. H. Reif, Symmetric complementation, Proc. of the 14th STOC, pp. 201–214, 1982.

    Google Scholar 

  49. R. Szelepcsenyi, The method of forcing for nondeterministic automata, Bull. of the European Ass. of Theoretical Computer Science, 33, pp. 96–100, 1987.

    MATH  Google Scholar 

  50. W. Savitch, Relashionships between nondeterministic and deterministic tape complexities, Journal of Computer Systems and Sciences, 4, pp. 177–192, 1970.

    MATH  MathSciNet  Google Scholar 

  51. S. Skyum and L. Valiant, A complexity theory based on Boolean algebra, Proc. of the 22nd FOCS, pp. 244–253, 1981.

    Google Scholar 

  52. 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.

    Article  MATH  MathSciNet  Google Scholar 

  53. A. C. Yao, Some complexity questions related to distributive computing, Proc. of the 11th STOC, pp. 209–213, 1979.

    Google Scholar 

  54. A. C. Yao, Private communication.

    Google Scholar 

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Authors and Affiliations

  1. Hebrew University and Princeton University, USA

    Avi Wigderson

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  1. Avi Wigderson
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Ivan M. Havel Václav Koubek

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© 1992 Springer-Verlag Berlin Heidelberg

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Wigderson, A. (1992). The complexity of graph connectivity. In: Havel, I.M., Koubek, V. (eds) Mathematical Foundations of Computer Science 1992. MFCS 1992. Lecture Notes in Computer Science, vol 629. Springer, Berlin, Heidelberg . https://doi.org/10.1007/3-540-55808-X_10

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  • DOI: https://doi.org/10.1007/3-540-55808-X_10

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55808-8

  • Online ISBN: 978-3-540-47291-9

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