  We give new partially-dynamic algorithms for the all-pairs shortest paths
problem in weighted directed graphs. Most importantly, we give a new
deterministic incremental algorithm for the problem that handles updates in
$\widetilde{O}(mn^{4/3}\log{W}/\epsilon)$ total time (where the edge weights
are from $[1,W]$) and explicitly maintains a $(1+\epsilon)$-approximate
distance matrix. For a fixed $\epsilon>0$, this is the first deterministic
partially dynamic algorithm for all-pairs shortest paths in directed graphs,
whose update time is $o(n^2)$ regardless of the number of edges. Furthermore,
we also show how to improve the state-of-the-art partially dynamic randomized
algorithms for all-pairs shortest paths [Baswana et al. STOC'02, Bernstein
STOC'13] from Monte Carlo randomized to Las Vegas randomized without increasing
the running time bounds (with respect to the $\widetilde{O}(\cdot)$ notation).
  Our results are obtained by giving new algorithms for the problem of
dynamically maintaining hubs, that is a set of $\widetilde{O}(n/d)$ vertices
which hit a shortest path between each pair of vertices, provided it has
hop-length $\Omega(d)$. We give new subquadratic deterministic and Las Vegas
algorithms for maintenance of hubs under either edge insertions or deletions.
