  We present parameterized streaming algorithms for the graph matching problem
in both the dynamic and the insert-only models. For the dynamic streaming
model, we present a one-pass algorithm that, with high probability, computes a
maximum-weight $k$-matching of a weighted graph in $\tilde{O}(Wk^2)$ space and
that has $\tilde{O}(1)$ update time, where $W$ is the number of distinct edge
weights and the notation $\tilde{O}()$ hides a poly-logarithmic factor in the
input size. For the insert-only streaming model, we present a one-pass
algorithm that runs in $O(k^2)$ space and has $O(1)$ update time, and that,
with high probability, computes a maximum-weight $k$-matching of a weighted
graph. The space complexity and the update-time complexity achieved by our
algorithms for unweighted $k$-matching in the dynamic model and for weighted
$k$-matching in the insert-only model are optimal.
  A notable contribution of this paper is that the presented algorithms {\it do
not} rely on the apriori knowledge/promise that the cardinality of \emph{every}
maximum-weight matching of the input graph is upper bounded by the parameter
$k$. This promise has been a critical condition in previous works, and lifting
it required the development of new tools and techniques.
