## Abstract

The Planar Graph Metric Compression Problem is to compactly encode the distances among k nodes in a planar graph of size n. Two naïve solutions are to store the graph using O(n) bits, or to explicitly store the distance matrix with O(k2 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 weighted planar graphs, but leave a large gap for unweighted planar graphs. For example, when k/p n, the upper bound is O(n) and their constructions imply an (n3/4) lower bound. This gap is directly related to other major open questions in labeling schemes, dynamic algorithms, and compact routing. Our main result is a new compression of the planar graph metric into ~O(min(k2; p k/n)) bits, which is optimal up to log factors. Our data structure circumvents an (k2) lower bound of Krauthgamer, Nguyen, and Zondiner [SIDMA'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 Subset Distance Oracle for planar graphs with ~O( √k/n) space, and ~O(n3/4) query time. Our work carries strong messages to related fields. In particular, the famous O(n1/2) vs. (n1/3) gap for distance labeling schemes in planar graphs cannot be resolved with the current lower bound techniques. On the positive side, we introduce the powerful tool of unit-monge to planar graph algorithms.

Original language | English |
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Title of host publication | 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 |

Editors | Artur Czumaj |

Publisher | Association for Computing Machinery |

Pages | 530-549 |

Number of pages | 20 |

ISBN (Electronic) | 9781611975031 |

DOIs | |

State | Published - 2018 |

Event | 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 - New Orleans, United States Duration: 7 Jan 2018 → 10 Jan 2018 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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### Conference

Conference | 29th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2018 |
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Country/Territory | United States |

City | New Orleans |

Period | 7/01/18 → 10/01/18 |

### Bibliographical note

Funding Information:‡IDCHerzliya, EfiAraziSchoolofComputerScience, smozes@idc.ac.il. Partially supported by the Israel Science Foun-dationgrants 794/13and592/17.

Funding Information:

§Universityof Haifa, Departmentof ComputerScience, oren@cs.haifa.ac.il. Partially supported by the Israel Science Foundationgrants 794/13and592/17.

Publisher Copyright:

© Copyright 2018 by SIAM.

## ASJC Scopus subject areas

- Software
- Mathematics (all)