Biased random walks

Abstract

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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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). https://doi.org/10.1007/BF01300124

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Mathematics Subject Classification (1991)

  • 60 J 15
  • 90 C 40
  • 68 R 10