  We introduce the notion of {\em Distance Restricted Manipulation}, where
colluding manipulator(s) need to compute if there exist votes which make their
preferred alternative win the election when their knowledge about the others'
votes is a little inaccurate. We use the Kendall-Tau distance to model the
manipulators' confidence in the non-manipulators' votes. To this end, we study
this problem in two settings - one where the manipulators need to compute a
manipulating vote that succeeds irrespective of perturbations in others' votes
({\em Distance Restricted Strong Manipulation}), and the second where the
manipulators need to compute a manipulating vote that succeeds for at least one
possible vote profile of the others ({\em Distance Restricted Weak
Manipulation}). We show that {\em Distance Restricted Strong Manipulation}
admits polynomial-time algorithms for every scoring rule, maximin, Bucklin, and
simplified Bucklin voting rules for a single manipulator, and for the
$k$-approval rule for any number of manipulators, but becomes intractable for
the Copeland$^\alpha$ voting rule for every $\alpha\in[0,1]$ even for a single
manipulator. In contrast, {\em Distance Restricted Weak Manipulation} is
intractable for almost all the common voting rules, with the exception of the
plurality rule. For a constant number of alternatives, we show that both the
problems are polynomial-time solvable for every anonymous and efficient voting
rule.
