 
Summary: Biased Random Walks
Yossi Azar \Lambda Andrei Z. Broder y Anna R. Karlin z
Nathan Linial x Steven Phillips 
Abstract
How much can an imperfect source of randomness affect an algo
rithm? We examine several simple questions of this type concerning
the longterm behavior of a random walk on a finite graph. In our
setup, at each step of the random walk a ``controller'' can, with a cer
tain 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 longterm
expected benefit. We derive tight bounds on the maximum of this ob
jective function over all controller's strategies, and present polynomial
time algorithms for computing the optimal controller strategy.
1 Introduction
Ever since the introduction of randomness into computing, people have been
studying how imperfections in the sources of randomness affect the outcome
\Lambda Department of Computer Science, Tel Aviv University, Israel. This research was
supported in part by the Alon Fellowship and the Israel Science Foundation administered
