  We give an almost-linear time algorithm for the Steiner connectivity
augmentation problem: given an undirected graph, find a smallest (or minimum
weight) set of edges whose addition makes a given set of terminals
$\tau$-connected (for any given $\tau > 0$). The running time of our algorithm
is dominated by polylogarithmic calls to any maximum flow subroutine; using the
recent almost-linear time maximum flow algorithm (Chen et al., FOCS 2022), we
get an almost-linear running time for our algorithm as well. This is tight up
to the polylogarithmic factor even for just two terminals. Prior to our work,
an almost-linear (in fact, near-linear) running time was known only for the
special case of global connectivity augmentation, i.e., when all vertices are
terminals (Cen et al., STOC 2022).
  We also extend our algorithm to the closely related Steiner splitting-off
problem, where the edges incident on a vertex have to be {\em split-off} while
maintaining the (Steiner) connectivity of a given set of terminals. Prior to
our work, a nearly-linear time algorithm was known only for the special case of
global connectivity (Cen et al., STOC 2022). The only known generalization
beyond global connectivity was to preserve all pairwise connectivities using a
much slower algorithm that makes $n$ calls to an all-pairs maximum flow (or
Gomory-Hu tree) subroutine (Lau and Yung, SICOMP 2013), as against polylog(n)
calls to a (single-pair) maximum flow subroutine in this work.
