Biased random walks

Authors:
Yossi Azar;Andrei Z. Broder;Anna R. Karlin;Nathan Linial;Steven Phillips
Affiliations:
Digital Equipment Corporation Systems Research Center, 130 Lytton Avenue, Palo Alto, CA;Digital Equipment Corporation Systems Research Center, 130 Lytton Avenue, Palo Alto, CA;Digital Equipment Corporation Systems Research Center, 130 Lytton Avenue, Palo Alto, CA;The Hebrew University of Jerusalem;Computer Science Department, Stanford University, Stanford, CA
Venue:
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Year:
1992

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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, 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.