Let G = (V,E) be an undirected unweighted planar graph. Let S = {s_1,…,s_k} be the vertices of some face in G and let T ⊆ V be an arbitrary set of vertices. The Okamura-Seymour metric compression problem asks to compactly encode the S-to-T distances. Consider a vector storing the distances from an arbitrary vertex v to all vertices S = {s_1,…,s_k} 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 p_#, is only O(k³). This resulted in a simple Õ(min{k⁴+|T|, k⋅|T|}) space compression of the Okamura-Seymour metric. We give an alternative proof of the p_# = O(k³) 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(p_#+k+|T|) space compression of the Okamura-Seymour metric, thus improving the compression of Li and Parter to Õ(min{k³+|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 p_# = Θ(k²) for the family of Halin graphs, whereas the VC-dimension argument is limited to showing p_# = O(k³).
@InProceedings{mozes_et_al:LIPIcs.ISAAC.2022.27, author = {Mozes, Shay and Wallheimer, Nathan and Weimann, Oren}, title = {{Improved Compression of the Okamura-Seymour Metric}}, booktitle = {33rd International Symposium on Algorithms and Computation (ISAAC 2022)}, pages = {27:1--27:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-258-7}, ISSN = {1868-8969}, year = {2022}, volume = {248}, editor = {Bae, Sang Won and Park, Heejin}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2022.27}, URN = {urn:nbn:de:0030-drops-173123}, doi = {10.4230/LIPIcs.ISAAC.2022.27}, annote = {Keywords: Shortest paths, planar graphs, metric compression, distance oracles} }
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