  Let $G=(V,E)$ be an undirected unweighted planar graph. Consider a vector
storing the distances from an arbitrary vertex $v$ to all vertices $S = \{ s_1
, s_2 , \ldots , s_k \}$ of a single face in their cyclic order. The pattern of
$v$ is obtained by taking the difference between every pair of consecutive
values of this vector. In STOC'19, Li and Parter used a VC-dimension argument
to show that in planar graphs, the number of distinct patterns, denoted $x$, is
only $O(k^3)$. This resulted in a simple compression scheme requiring $\tilde
O(\min \{ k^4+|T|, k\cdot |T|\})$ space to encode the distances between $S$ and
a subset of terminal vertices $T \subseteq V$. This is known as the
Okamura-Seymour metric compression problem.
  We give an alternative proof of the $x=O(k^3)$ bound that exploits planarity
beyond the VC-dimension argument. Namely, our proof relies on cut-cycle
duality, as well as on the fact that distances among vertices of $S$ are
bounded by $k$. Our method implies the following:
  (1) An $\tilde{O}(x+k+|T|)$ space compression of the Okamura-Seymour metric,
thus improving the compression of Li and Parter to $\tilde O(\min \{k^3+|T|,k
\cdot |T| \})$.
  (2) An optimal $\tilde{O}(k+|T|)$ space compression of the Okamura-Seymour
metric, in the case where the vertices of $T$ induce a connected component in
$G$.
  (3) A tight bound of $x = \Theta(k^2)$ for the family of Halin graphs,
whereas the VC-dimension argument is limited to showing $x=O(k^3)$.
