  In this paper, we present new algorithms for approximating All-Pairs Shortest
Paths (APSP) in the Congested Clique model. We present randomized algorithms
for weighted undirected graphs.
  Our first contribution is an $O(1)$-approximate APSP algorithm taking just
$O(\log \log \log n)$ rounds. Prior to our work, the fastest algorithms that
give an $O(1)$-approximation for APSP take $\operatorname{poly}(\log{n})$
rounds in weighted undirected graphs, and $\operatorname{poly}(\log \log n)$
rounds in unweighted undirected graphs.
  If we terminate the execution of the algorithm early, we obtain an
$O(t)$-round algorithm that yields an $O \big( (\log n)^{1/2^t} \big) $
distance approximation for a parameter $t$. The trade-off between $t$ and the
approximation quality provides flexibility for different scenarios, allowing
the algorithm to adapt to specific requirements. In particular, we can get an
$O \big( (\log n)^{1/2^t} \big) $-approximation for any constant $t$ in
$O(1)$-rounds. Such result was previously known only for the special case that
$t=0$.
  A key ingredient in our algorithm is a lemma that allows to improve an
$O(a)$-approximation for APSP to an $O(\sqrt{a})$-approximation for APSP in
$O(1)$ rounds. To prove the lemma, we develop several new tools, including
$O(1)$-round algorithms for computing the $k$ closest nodes, a certain type of
hopset, and skeleton graphs.
