  In this paper we investigate the top-$k$-selection problem, i.e. determine
the largest, second largest, ..., and the $k$-th largest elements, in the
dynamic data model. In this model the order of elements evolves dynamically
over time. In each time step the algorithm can only probe the changes of data
by comparing a pair of elements. Previously only two special cases were
studied[2]: finding the largest element and the median; and sorting all
elements. This paper systematically deals with $k\in [n]$ and solves the
problem almost completely. Specifically, we identify a critical point $k^*$
such that the top-$k$-selection problem can be solved error-free with
probability $1-o(1)$ if and only if $k=o(k^*)$. A lower bound of the error when
$k=\Omega(k^*)$ is also determined, which actually is tight under some
condition. On the other hand, it is shown that the top-$k$-set problem, which
means finding the largest $k$ elements without sorting them, can be solved
error-free for all $k\in [n]$. Additionally, we extend the dynamic data model
and show that most of these results still hold.
