Biased random walks


How much can an imperfect source of randomness affect an algorithm? We examine several simple questions of this type concerning the long-term behavior of a random walk on a finite graph. In our setup, at each step of the random walk a “controller” can, with a certain small probability, fix the next step, thus introducing a bias. We analyze the extent to which the bias can affect the limit behavior of the walk. The controller is assumed to associate a real, nonnegative, “benefit” with each state, and to strive to maximize the long-term expected benefit. We derive tight bounds on the maximum of this objective function over all controller's strategies, and present polynomial time algorithms for computing the optimal controller strategy.

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  1. [1]

    M. Ajtai, J. Komlós, andE. Szemerédi: Deterministic Simulation in LOGSPACE,19th Annual ACM STOC, 1987, 132–140.

  2. [2]

    M. Ajtai, andN. Linial: The influence of large coalitions,Combinatorica,13 (1993) 129–145.

    Google Scholar 

  3. [3]

    N. Alon, andM. Naor: Coin-flipping Games Immune Against Linear-sized Coalitions,SIAM J. Comput. 22 (1993), 403–417.

    Google Scholar 

  4. [4]

    N. Alon, andM. O. Rabin: Biased Coins and Randomized Algorithms, inRandomness and Computation (S. Micali ed.) Advances in Computing Research, Vol. 5, 499–507.

  5. [5]

    Y. Azar, A. Z. Broder, A. R. Karlin, N. Linial, andS. J. Phillips: Biased Random Walks, in24th Annual ACM STOC, 1992, 1–9.

  6. [6]

    M. Ben-Or, andN. Linial: Collective coin flipping, inRandomness and Computation (S. Micali ed.) Academic Press, New York, 1990, 91–115.

    Google Scholar 

  7. [7]

    M. Ben-Or, N. Linial, andM. Saks: Collective coin flipping and other models of imperfect randomness, inColloq. Math. Soc. János Bolyai no. 52, Combinatorics Eger, 1987, 75–112.

  8. [8]

    A. Cohen, andA. Wigderson: Dispersers, Deterministic Amplification, and Weak Random Sources,30th Annual IEEE FOCS, 1989, 14–19.

  9. [9]

    C. Derman:Finite State Markov Decision Processes, Academic Press, New York, 1970.

    Google Scholar 

  10. [10]

    O. Goldreich, S. Goldwasser, andN. Linial: Fault-tolerant Computation in the Full Information Model,32nd Annual IEEE FOCS, 1991, 447–457.

  11. [11]

    E. J. Hinch:Perturbation methods, Cambridge University Press, 1991.

  12. [12]

    R. Impagliazzo, andD. Zuckerman: How to Recycle Random Bits,30th Annual IEEE FOCS, 1989, 248–253.

  13. [13]

    T. Kato:Perturbation theory for linear operators, New York, Springer Verlag, 1982.

    Google Scholar 

  14. [14]

    D. Lichtenstein, N. Linial, andM. Saks: Some extremal problems arising from discrete control processes,Combinatorica,9 (1989), 269–287.

    Google Scholar 

  15. [15]

    M. Metropolis, A. Rosenbluth, M. Rosenbluth, A. Teller, andM. Teller: Equation of state calculations by fast computing machines,Journal of Chemical Physics 21 (1953), 1087–1092.

    Google Scholar 

  16. [16]

    F. Rellich:Perturbation theory of eigenvalue problems, New York, Gordon and Breach, 1969.

    Google Scholar 

  17. [17]

    M. Santha, andU. Vazirani: Generating Quasi-Random Sequences from Semirandom Sources,J. Comput. System Sci. 33 (1986), 75–87.

    Google Scholar 

  18. [18]

    U. Vazirani: Efficiency Considerations in Using Slightly-Random Sources,19th Annual ACM STOC, 1987, 160–168.

  19. [19]

    U. Vazirani: Randomness, Adversaries and Computation, Ph.D. Thesis, University of California, Berkeley, 1986.

    Google Scholar 

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Additional information

This research was supported in part by the Alon Fellowship and the Israel Science Foundation administered by the Israel Academy of Sciences. This work was supported in part by a grant from the Israeli Academy of Sciences. A portion of this work was done while the author was visiting DEC Systems Research Center.

A portion of this work was done while the author was in the Computer Science Department at Stanford and while visiting DEC Systems Research Center. Partially supported by NSF Grant CCR-9010517 and an OTL grant.

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Azar, Y., Broder, A.Z., Karlin, A.R. et al. Biased random walks. Combinatorica 16, 1–18 (1996).

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Mathematics Subject Classification (1991)

  • 60 J 15
  • 90 C 40
  • 68 R 10