Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
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)
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)
Chaves, L.M., de Souza, D.J.: Waiting Time for a Run of N Success in Bernoulli Sequences. Rev. Bras. Biom (2007)
Chung, F.R.K.: Laplacians and the cheeger inequality for directed graphs. Annals of Combinatorics 9, 1–19 (2005)
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)
Feige, U.: A tight upper bound on the cover time for random walks on graphs. Random Structures and Algorithms 6, 51–54 (1995)
Garvin, B., Stolee, D., Tewari, R., Vinodchandran, N.V.: ReachFewL = ReachUL. Electronic Colloquium on Computational Complexity (ECCC) 18, 60 (2011)
Hartmanis, J., Immerman, N., Mahaney, S.: One-way Log-tape Reductions. In: Proc. IEEE FOCS, pp. 65–71 (1978)
Kazarinoff, N.D.: Analytic inequalities. Holt, Rinehart and Winston, New York (1961)
Lewis, H.R., Papadimitriou, C.H.: Symmetric space-bounded computation. Theoretical Computer Science. pp.161-187 (1982)
Li, Y., Zhang, Z.-L.: Digraph Laplacian and the Degree of Asymmetry. Invited for submission to a Special Issue of Internet Mathematics (2011)
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)
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)
Reingold, O.: Undirected ST-connectivity in log-space. In: STOC 2005, pp. 376–385 (2005)
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)
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)
Savitch, W.J.: Relationships Between Nondeterministic and Deterministic Tape Complexityies. J. Comput. Syst. Sci. 4(2), 177–192 (1970)
Saks, M.E., Zhou, S.: BPHSpace(S) ⊆ DSPace(S3/2). J. Comput. Syst. Sci. 58(2), 36–403 (1999)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cheng, WJ., Cox, J., Zachos, S. (2013). Random Walks on Some Basic Classes of Digraphs. In: Liu, Z., Woodcock, J., Zhu, H. (eds) Theoretical Aspects of Computing – ICTAC 2013. ICTAC 2013. Lecture Notes in Computer Science, vol 8049. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39718-9_8
Download citation
DOI: https://doi.org/10.1007/978-3-642-39718-9_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39717-2
Online ISBN: 978-3-642-39718-9
eBook Packages: Computer ScienceComputer Science (R0)