  We develop a new approach for distributed distance computation in planar
graphs that is based on a variant of the metric compression problem recently
introduced by Abboud et al. [SODA'18]. One of our key technical contributions
is in providing a compression scheme that encodes all $S \times T$ distances
using $\widetilde{O}(|S|\cdot poly(D)+|T|)$ bits for unweighted graphs with
diameter $D$. This significantly improves the state of the art of
$\widetilde{O}(|S|\cdot 2^{D}+|T| \cdot D)$ bits. We also consider an
approximate version of the problem for \emph{weighted} graphs, where the goal
is to encode $(1+\epsilon)$ approximation of the $S \times T$ distances. At the
heart of this compact compression scheme lies a VC-dimension type argument on
planar graphs. This efficient compression scheme leads to several improvements
and simplifications in the setting of diameter computation, most notably in the
distributed setting:
  - There is an $\widetilde{O}(D^5)$-round randomized distributed algorithm for
computing the diameter in planar graphs, w.h.p.
  - There is an $\widetilde{O}(D^3)+ poly(\log n/\epsilon)\cdot D^2$-round
randomized distributed algorithm for computing an $(1+\epsilon)$ approximation
of the diameter in weighted graphs with polynomially bounded weights, w.h.p.
  No sublinear round algorithms were known for these problems before. These
distributed constructions are based on a new recursive graph decomposition that
preserves the (unweighted) diameter of each of the subgraphs up to a
logarithmic term. Using this decomposition, we also get an \emph{exact} SSSP
tree computation within $\widetilde{O}(D^2)$ rounds.
