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Abstract

We introduce a “derandomized” analogue of graph squaring. This operation increases the connectivity of the graph (as measured by the second eigenvalue) almost as well as squaring the graph does, yet only increases the degree of the graph by a constant factor, instead of squaring the degree.

One application of this product is an alternative proof of Reingold’s recent breakthrough result that S-T Connectivity in Undirected Graphs can be solved in deterministic logspace.

Keywords

Undirected Graph Regular Graph Input Graph Edge Label Pseudorandom Generator 
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.

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References

  1. [AKL+]
    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 1979, San Juan, Puerto Rico, pp. 218–223. IEEE, New York (1979)CrossRefGoogle Scholar
  2. [AFWZ]
    Alon, N., Feige, U., Wigderson, A., Zuckerman, D.: Derandomized graph products. Comput. Complexity 5(1), 60–75 (1995)zbMATHCrossRefMathSciNetGoogle Scholar
  3. [AS]
    Alon, N., Sudakov, B.: Bipartite subgraphs and the smallest eigenvalue. Combin. Probab. Comput. 9(1), 1–12 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  4. [GG]
    Gabber, O., Galil, Z.: Explicit Constructions of Linear-Sized Superconcentrators. J. Comput. Syst. Sci. 22(3), 407–420 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  5. [HW]
    Hoory, S., Wigderson, A.: Universal Traversal Sequences for Expander Graphs. Inf. Process. Lett. 46(2), 67–69 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  6. [INW]
    Impagliazzo, R., Nisan, N., Wigderson, A.: Pseudorandomness for Network Algorithms. In: Proceedings of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, Montréal, Québec, Canada, May 23-25, pp. 356–364 (1994)Google Scholar
  7. [MR]
    Martin, R.A., Randall, D.: Sampling Adsorbing Staircase Walks Using a New Markov Chain Decomposition Method. In: Proceedings of the 41st Annual Symposium on Foundations of Computer Science, Redondo Beach, CA, October 17–19, pp. 492–502. IEEE, Los Alamitos (2000)CrossRefGoogle Scholar
  8. [Mih]
    Mihail, M.: Conductance and convergence of markov chains: a combinatorial treatment of expanders. In: Proc. of the 37th Conf. on Foundations of Computer Science, pp. 526–531 (1989)Google Scholar
  9. [RTV]
    Reingold, Trevisan, and Vadhan. Pseudorandom Walks in Biregular Graphs and the RL vs. L Problem. In: ECCCTR: Electronic Colloquium on Computational Complexity, technical reports (2005)Google Scholar
  10. [Rei]
    Reingold, O.: Undirected ST-connectivity in log-space. In: STOC 2005: Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, pp. 376–385. ACM Press, New York (2005)CrossRefGoogle Scholar
  11. [RTV]
    Reingold, O., Trevisan, L., Vadhan, S.: Pseudorandom Walks in Biregular Graphs and the RL vs. L Problem. Electronic Colloquium on Computational Complexity Technical Report TR05-022 (February 2005), http://www.eccc.uni-trier.de/eccc
  12. [RVW]
    Reingold, O., Vadhan, S., Wigderson, A.: Entropy waves, the zig-zag graph product, and new constant-degree expanders. Ann. of Math (2) 155(1), 157–187 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  13. [Sav]
    Savitch, W.J.: Relationships between nondeterministic and deterministic tape complexities. J. Comput. System. Sci. 4, 177–192 (1970)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Eyal Rozenman
    • 1
  • Salil Vadhan
    • 2
  1. 1.Division of Engineering & Applied SciencesHarvard UniversityCambridge
  2. 2.Department of Computer Science & Applied MathematicsWeizmann InstituteRehovotIsrael

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