Improved Compression of the Okamura-Seymour Metric
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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 , … , 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 Õ(min{ k^4+|T|, k· |T|}) space to encode the distances between S and a subset of terminal vertices T ⊆ 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 Õ(x+k+|T|) space compression of the Okamura-Seymour metric, thus improving the compression of Li and Parter to Õ(min{k^3+|T|,k · |T| }). (2) An optimal Õ(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 = Θ(k^2) for the family of Halin graphs, whereas the VC-dimension argument is limited to showing x=O(k^3).
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