Random Walks on Some Basic Classes of Digraphs

  • Wen-Ju Cheng
  • Jim Cox
  • Stathis Zachos
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8049)


Reingold has shown that L=SL, that s-t connectivity in a poly-mixing digraph is complete for promise-RL, and that s-t connectivity for a poly-mixing out-regular digraph with known stationary distribution is in L. However, little work has been done on identifying structural properties of digraphs that effect cover times. We examine the complexity of random walks on a basic parameterized family of unbalanced digraphs called Strong Chains (which model weakly symmetric computation), and a special family of Strong Chains called Harps. We show that the worst case hitting times of Strong Chain families vary smoothly with the number of asymmetric vertices and identify the necessary condition for non-polynomial cover time. This analysis also yields bounds on the cover times of general digraphs. Our goal is to use these structural properties to develop space efficient digraph modification for randomized search and to develop derandomized search strategies for digraph families.


complexity space bounded complexity classes random walks digraph search reachability strong connectivity symmetric computation RL NL BPL 


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  1. [AKLLR]
    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 IEEE Symp. on Found. of Computer Science (FOCS 1979), pp. 218–223 (1979)Google Scholar
  2. [ATWZ]
    Armoni, R., Ta-Shma, A., Wigderson, A., Zhou, S.: An O(log(n)4/3) space algorithm for (s, t) connectivity in undirected graphs. J. ACM 47(2), 294–311 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  3. [CS]
    Chaves, L.M., de Souza, D.J.: Waiting Time for a Run of N Success in Bernoulli Sequences. Rev. Bras. Biom (2007)Google Scholar
  4. [Chung]
    Chung, F.R.K.: Laplacians and the cheeger inequality for directed graphs. Annals of Combinatorics 9, 1–19 (2005)Google Scholar
  5. [CRV]
    Chung, K.-M., Reingold, O., Vadhan, S.: S-T Connectivity on Digraphs with a Known Stationary Distribution. ACM Transactions on Algorithms 7(3), Article 30 (2011)Google Scholar
  6. [Feige]
    Feige, U.: A tight upper bound on the cover time for random walks on graphs. Random Structures and Algorithms 6, 51–54 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  7. [GSTV]
    Garvin, B., Stolee, D., Tewari, R., Vinodchandran, N.V.: ReachFewL = ReachUL.  Electronic Colloquium on Computational Complexity (ECCC) 18, 60 (2011)Google Scholar
  8. [HIM]
    Hartmanis, J., Immerman, N., Mahaney, S.: One-way Log-tape Reductions. In: Proc. IEEE FOCS, pp. 65–71 (1978)Google Scholar
  9. [Kazarinoff]
    Kazarinoff, N.D.: Analytic inequalities. Holt, Rinehart and Winston, New York (1961)zbMATHGoogle Scholar
  10. [LP]
    Lewis, H.R., Papadimitriou, C.H.: Symmetric space-bounded computation. Theoretical Computer Science. pp.161-187 (1982)Google Scholar
  11. [LZ]
    Li, Y., Zhang, Z.-L.: Digraph Laplacian and the Degree of Asymmetry. Invited for submission to a Special Issue of Internet Mathematics (2011)Google Scholar
  12. [NSW]
    Nisan, N., Szemeredi, E., Wigderson, A.: Undirected connectivity in O(log1.5n) space. In: 33rd Annual Symposium on Foundations of Computer Science, Pittsburgh, Pa, October 1992. IEEE (1992)Google Scholar
  13. [PRV]
    Pavan, A., Tewari, R., Vinodchandran, N.V.: On the Power of Unambiguity in Logspace. Technical Report TR10-009, Electronic Colloquium on Computational Complexity, To appear in Computational Complexity (2010)Google Scholar
  14. [Reingold]
    Reingold, O.: Undirected ST-connectivity in log-space. In: STOC 2005, pp. 376–385 (2005)Google Scholar
  15. [RTV]
    Reingold, O., Trevisan, L., Vadhan, S.P.: Pseudorandom walks on regular digraphs and the RL vs. L Problem. In: STOC 2006, pp. 457–466 (2006)Google Scholar
  16. [RV]
    Rozenman, E., Vadhan, S.: Derandomized Squaring of Graphs. In: Chekuri, C., Jansen, K., Rolim, J.D.P., Trevisan, L. (eds.) APPROX and RANDOM 2005. LNCS, vol. 3624, pp. 436–447. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  17. [Savitch]
    Savitch, W.J.: Relationships Between Nondeterministic and Deterministic Tape Complexityies. J. Comput. Syst. Sci. 4(2), 177–192 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  18. [SZ]
    Saks, M.E., Zhou, S.: BPHSpace(S) ⊆ DSPace(S3/2). J. Comput. Syst. Sci. 58(2), 36–403 (1999)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Wen-Ju Cheng
    • 1
  • Jim Cox
    • 1
  • Stathis Zachos
    • 1
    • 2
  1. 1.Computer Science DepartmentCUNY Graduate CenterNew YorkUSA
  2. 2.CS Dept, School of ECENTUAGreece

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