## Abstract

Let G = (V, E) be an undirected unweighted planar graph. Let S = {s1, . . ., 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 = {s1, . . ., 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(k^{3}). This resulted in a simple Õ(min{k^{4} + |T|, k · |T|}) space compression of the Okamura-Seymour metric. We give an alternative proof of the p_{#} = 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(p_{#} + 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 p_{#} = Θ(k^{2}) for the family of Halin graphs, whereas the VC-dimension argument is limited to showing p_{#} = O(k^{3}).

Original language | English |
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Title of host publication | 33rd International Symposium on Algorithms and Computation, ISAAC 2022 |

Editors | Sang Won Bae, Heejin Park |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959772587 |

DOIs | |

State | Published - 1 Dec 2022 |

Event | 33rd International Symposium on Algorithms and Computation, ISAAC 2022 - Virtual, Online, Korea, Republic of Duration: 19 Dec 2022 → 21 Dec 2022 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 248 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 33rd International Symposium on Algorithms and Computation, ISAAC 2022 |
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Country/Territory | Korea, Republic of |

City | Virtual, Online |

Period | 19/12/22 → 21/12/22 |

### Bibliographical note

Publisher Copyright:© Shay Mozes, Nathan Wallheimer, and Oren Weimann.

## Keywords

- Shortest paths
- distance oracles
- metric compression
- planar graphs

## ASJC Scopus subject areas

- Software