  We study approximate distributed solutions to the weighted {\it
all-pairs-shortest-paths} (APSP) problem in the CONGEST model. We obtain the
following results.
  $1.$ A deterministic $(1+o(1))$-approximation to APSP in $\tilde{O}(n)$
rounds. This improves over the best previously known algorithm, by both
derandomizing it and by reducing the running time by a $\Theta(\log n)$ factor.
  In many cases, routing schemes involve relabeling, i.e., assigning new names
to nodes and require that these names are used in distance and routing queries.
It is known that relabeling is necessary to achieve running times of $o(n/\log
n)$. In the relabeling model, we obtain the following results.
  $2.$ A randomized $O(k)$-approximation to APSP, for any integer $k>1$,
running in $\tilde{O}(n^{1/2+1/k}+D)$ rounds, where $D$ is the hop diameter of
the network. This algorithm simplifies the best previously known result and
reduces its approximation ratio from $O(k\log k)$ to $O(k)$. Also, the new
algorithm uses uses labels of asymptotically optimal size, namely $O(\log n)$
bits.
  $3.$ A randomized $O(k)$-approximation to APSP, for any integer $k>1$,
running in time $\tilde{O}((nD)^{1/2}\cdot n^{1/k}+D)$ and producing {\it
compact routing tables} of size $\tilde{O}(n^{1/k})$. The node lables consist
of $O(k\log n)$ bits. This improves on the approximation ratio of $\Theta(k^2)$
for tables of that size achieved by the best previously known algorithm, which
terminates faster, in $\tilde{O}(n^{1/2+1/k}+D)$ rounds.
