  The fragile complexity of a comparison-based algorithm is $f(n)$ if each
input element participates in $O(f(n))$ comparisons. In this paper, we explore
the fragile complexity of algorithms adaptive to various restrictions on the
input, i.e., algorithms with a fragile complexity parameterized by a quantity
other than the input size n. We show that searching for the predecessor in a
sorted array has fragile complexity ${\Theta}(\log k)$, where $k$ is the rank
of the query element, both in a randomized and a deterministic setting. For
predecessor searches, we also show how to optimally reduce the amortized
fragile complexity of the elements in the array. We also prove the following
results: Selecting the $k$-th smallest element has expected fragile complexity
$O(\log \log k)$ for the element selected. Deterministically finding the
minimum element has fragile complexity ${\Theta}(\log(Inv))$ and
${\Theta}(\log(Runs))$, where $Inv$ is the number of inversions in a sequence
and $Runs$ is the number of increasing runs in a sequence. Deterministically
finding the median has fragile complexity $O(\log(Runs) + \log \log n)$ and
${\Theta}(\log(Inv))$. Deterministic sorting has fragile complexity
${\Theta}(\log(Inv))$ but it has fragile complexity ${\Theta}(\log n)$
regardless of the number of runs.
