Improved Compression of the Okamura-Seymour Metric

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 \(x=O(k^3)\).