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Simulating Undirected st-Connectivity Algorithms on Uniform JAGs and NNJAGs

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Algorithms and Computation (ISAAC 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3827))

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Abstract

In a breakthrough result, Reingold [17] showed that the Undirected st-Connectivity problem can be solved in O(log n) space. The next major challenge in this direction is whether one can extend it to directed graphs, and thereby lowering the deterministic space complexity of \(\mathcal{RL}\) or \(\mathcal{NL}\). In this paper, we show that Reingold’s algorithm, the O(log4/3 n)-space algorithm by Armoni et al.[3] and the O(log3/2 n)-space algorithm by Nisan et al.[14] can all be carried out on the RAM-NNJAG model [15](a uniform version of the NNJAG model [16]). As there is a tight Ω(log2 n) space lower bound for the Directed st-Connectivity problem on the RAM-NNJAG model implied by [8], our result gives an obstruction to generalizing Reingold’s algorithm to the directed case.

The work described in this paper was fully supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China [CityU 1071/02E], an NSF Grant, USA (CCR-0208013) and the Natural Science Foundation of China (60223004, 60321002).

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References

  1. Aleliunas, R., Karp, R.M., Lipton, R.J., Lovász, L., Rackoff, C.: Random walks, universal traversal sequences, and the complexity of maze problems. In: 20th Annual Symposium on Foundations of Computer Science, San Juan, Puerto Rico, October 1979, pp. 218–223. IEEE, Los Alamitos (1979)

    Google Scholar 

  2. Alon, N., Roichman, Y.: Random Cayley graphs and and expanders. Random Structures and Algorithms 5(2), 271–284 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  3. Armoni, R., Ta-Shma, A., Wigderson, A., Zhou, S.: An O(log(n)4/3) space algorithm for (s,t) connectivity in undirected graphs. Journal of the ACM 47(2), 294–311 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  4. Barnes, G., Buss, J.F., Ruzzo, W.L., Schieber, B.: A sublinear space, polynomial time algorithm for directed s-t connectivity. SIAM Journal on Computing 27(5), 1273–1282 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  5. Berman, P., Simon, J.: Lower bounds on graph threading by probabilistic machines. In: 24th Annual Symposium on Foundations of Computer Science, Tucson, AZ, November 1983, pp. 304–311. IEEE, Los Alamitos (1983)

    Chapter  Google Scholar 

  6. Cook, S.A., Rackoff, C.W.: Space lower bounds for maze threadability on restricted machines. SIAM Journal on Computing 9(3), 636–652 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dinur, I., Reingold, O., Trevisan, L., Vadhan, S.: Finding paths in nonreversible markov chains. Technical Report TR05-022, Electronic Colloquium on Computational Complexity (2005)

    Google Scholar 

  8. Edmonds, J., Poon, C.K., Achlioptas, D.: Tight lower bounds for st-connectivity on the NNJAG model. SIAM Journal on Computing 28(6), 2257–2284 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Gopalan, P., Lipton, R., Mehta, A.: Randomized time-space tradeoffs for directed graph connectivity. In: Pandya, P.K., Radhakrishnan, J. (eds.) FSTTCS 2003. LNCS, vol. 2914, pp. 208–216. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  10. Immerman, N.: Nondeterministic space is closed under complementation. SIAM Journal on Computing 17(5), 935–938 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  11. Lewis, H.R., Papadimitriou, C.H.: Symmetric space-bounded computation. Theoretical Computer Science 19(2), 161–187 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  12. Lu, P., Zhang, J., Poon, C.K., Cai, J.: Simulating undirected st-oonnectivity algorithms on uniform JAGs and NNJAGs. Technical report (2005)

    Google Scholar 

  13. Nisan, N.: \({\it RL} \subseteq {\it SC}\). In: Proceedings of the Twenty Fourth Annual ACM Symposium on Theory of Computing, Victoria, B.C., Canada, May 1992, pp. 619–623 (1992)

    Google Scholar 

  14. Nisan, N., Szemerédi, E., Wigderson, A.: Undirected connectivity in O(log1.5 n) space. In: 33rd Annual Symposium on Foundations of Computer Science, Pittsburgh, PA, October 1992. IEEE, Los Alamitos (1992)

    Google Scholar 

  15. Poon, C.K.: On the Complexity of the ST-Connectivity Problem. PhD thesis, University of Toronto (1996)

    Google Scholar 

  16. Poon, C.K.: A space lower bound for st-connectivity on node-named JAGs. Theoretical Computer Science 237(1-2), 327–345 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  17. Reingold, O.: Undirected st-connectivity in log-space. In: Proceedings of the Thirty Seventh Annual ACM Symposium on Theory of Computing, pp. 376–385 (2005)

    Google Scholar 

  18. Reingold, O., Vadhan, S., Wigderson, A.: Entropy waves, the zig-zag graph product, and new constant-degree expanders. Annals of Mathematics 155(1) (2001)

    Google Scholar 

  19. Savitch, W.J.: Relationships between nondeterministic and deterministic tape complexities. Journal of Computer and System Sciences 4(2), 177–192 (1970)

    Article  MATH  MathSciNet  Google Scholar 

  20. Szelepcsényi, R.: The method of forcing for nondeterministic automata. Bulletin of the European Association for Theoretical Computer Science 33, 96–100 (1987)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. State Key Laboratory of Intelligent Technology and Systems, Department of Computer Science and Technology, Tsinghua University, Beijing, China

    Pinyan Lu

  2. Department of Physics, Tsinghua University, Beijing, China

    Jialin Zhang

  3. Department of Computer Science, City University of Hong Kong, Hong Kong, China

    Chung Keung Poon

  4. Computer Sciences Department, University of Wisconsin, Madison, WI, 53706, USA

    Jin-Yi Cai

  5. Tsinghua University, Beijing, China

    Jin-Yi Cai

Authors
  1. Pinyan Lu
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  2. Jialin Zhang
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  3. Chung Keung Poon
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  4. Jin-Yi Cai
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Editor information

Editors and Affiliations

  1. Department of Computer Science, City University of Hong Kong, Tat Chee Avenue, Kowloon , Hong Kong

    Xiaotie Deng

  2. Department of Computer Science, University of Texas at Dallas, EE/CS Building, 75083, Richardson, TX, USA

    Ding-Zhu Du

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© 2005 Springer-Verlag Berlin Heidelberg

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Lu, P., Zhang, J., Poon, C.K., Cai, JY. (2005). Simulating Undirected st-Connectivity Algorithms on Uniform JAGs and NNJAGs. In: Deng, X., Du, DZ. (eds) Algorithms and Computation. ISAAC 2005. Lecture Notes in Computer Science, vol 3827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11602613_77

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  • DOI: https://doi.org/10.1007/11602613_77

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-30935-2

  • Online ISBN: 978-3-540-32426-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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