  We study the $k$-connectivity augmentation problem ($k$-CAP) in the
single-pass streaming model. Given a $(k-1)$-edge connected graph $G=(V,E)$
that is stored in memory, and a stream of weighted edges $L$ with weights in
$\{0,1,\dots,W\}$, the goal is to choose a minimum weight subset $L'\subseteq
L$ such that $G'=(V,E\cup L')$ is $k$-edge connected. We give a
$(2+\epsilon)$-approximation algorithm for this problem which requires to store
$O(\epsilon^{-1} n\log n)$ words. Moreover, we show our result is tight: Any
algorithm with better than $2$-approximation for the problem requires
$\Omega(n^2)$ bits of space even when $k=2$. This establishes a gap between the
optimal approximation factor one can obtain in the streaming vs the offline
setting for $k$-CAP.
  We further consider a natural generalization to the fully streaming model
where both $E$ and $L$ arrive in the stream in an arbitrary order. We show that
this problem has a space lower bound that matches the best possible size of a
spanner of the same approximation ratio. Following this, we give improved
results for spanners on weighted graphs: We show a streaming algorithm that
finds a $(2t-1+\epsilon)$-approximate weighted spanner of size at most
$O(\epsilon^{-1} n^{1+1/t}\log n)$ for integer $t$, whereas the best prior
streaming algorithm for spanner on weighted graphs had size depending on $\log
W$. Using our spanner result, we provide an optimal $O(t)$-approximation for
$k$-CAP in the fully streaming model with $O(nk + n^{1+1/t})$ words of space.
  Finally we apply our results to network design problems such as Steiner tree
augmentation problem (STAP), $k$-edge connected spanning subgraph ($k$-ECSS),
and the general Survivable Network Design problem (SNDP). In particular, we
show a single-pass $O(t\log k)$-approximation for SNDP using $O(kn^{1+1/t})$
words of space, where $k$ is the maximum connectivity requirement.
