  We study the problem of computing a longest increasing subsequence in a
sequence $S$ of $n$ distinct elements in the presence of persistent comparison
errors. In this model, every comparison between two elements can return the
wrong result with some fixed (small) probability $ p $, and comparisons cannot
be repeated. Computing the longest increasing subsequence exactly is impossible
in this model, therefore, the objective is to identify a subsequence that (i)
is indeed increasing and (ii) has a length that approximates the length of the
longest increasing subsequence.
  We present asymptotically tight upper and lower bounds on both the
approximation factor and the running time. In particular, we present an
algorithm that computes an $O(\log n)$-approximation in time $O(n\log n)$, with
high probability. This approximation relies on the fact that that we can
approximately sort $n$ elements in $O(n\log n)$ time such that the maximum
dislocation of an element is at most $O(\log n)$. For the lower bounds, we
prove that (i) there is a set of sequences, such that on a sequence picked
randomly from this set every algorithm must return an $\Omega(\log
n)$-approximation with high probability, and (ii) any $O(\log n)$-approximation
algorithm for longest increasing subsequence requires $\Omega(n \log n)$
comparisons, even in the absence of errors.
