Near-Optimal Compression for the Planar Graph Metric

The Planar Graph Metric Compression Problem is to compactly encode the distances among $k$ nodes in a planar graph of size $n$. Two na\"ive solutions are to store the graph using $O(n)$ bits, or to explicitly store the distance matrix with $O(k^2 \log{n})$ bits. The only lower bounds are from the seminal work of Gavoille, Peleg, Prennes, and Raz [SODA'01], who rule out compressions into a polynomially smaller number of bits, for {\em weighted} planar graphs, but leave a large gap for unweighted planar graphs. For example, when $k=\sqrt{n}$, the upper bound is $O(n)$ and their constructions imply an $\Omega(n^{3/4})$ lower bound. This gap is directly related to other major open questions in labelling schemes, dynamic algorithms, and compact routing. Our main result is a new compression of the planar graph metric into $\tilde{O}(\min (k^2 , \sqrt{k\cdot n}))$ bits, which is optimal up to log factors. Our data structure breaks an $\Omega(k^2)$ lower bound of Krauthgamer, Nguyen, and Zondiner [SICOMP'14] for compression using minors, and the lower bound of Gavoille et al. for compression of weighted planar graphs. This is an unexpected and decisive proof that weights can make planar graphs inherently more complex. Moreover, we design a new {\em Subset Distance Oracle} for planar graphs with $\tilde O(\sqrt{k\cdot n})$ space, and $\tilde O(n^{3/4})$ query time. Our work carries strong messages to related fields. In particular, the famous $O(n^{1/2})$ vs. $\Omega(n^{1/3})$ gap for distance labelling schemes in planar graphs {\em cannot} be resolved with the current lower bound techniques.