## Abstract

We present an efficient construction of additively weighted Voronoi diagrams on planar graphs. Let G be a planar graph with n vertices and b sites that lie on a constant number of faces. We show how to preprocess G in Õ (nb2) time1 so that one can compute any additively weighted Voronoi diagram for these sites in Õ(b) time. We use this construction to compute the diameter of a directed planar graph with real arc lengths in O(n5/3) time. This improves the recent breakthrough result of Cabello (SODA'17), both by improving the running time (from Õ (n11/6)), and by providing a deterministic algorithm. It is in fact the first truly subquadratic deterministic algorithm for this problem. Our use of Voronoi diagrams to compute the diameter follows that of Cabello, but he used abstract Voronoi diagrams, which makes his diameter algorithm more involved, more expensive, and randomized. As in Cabello's work, our algorithm can also compute the Wiener index of a planar graph (i.e., the sum of all pairwise distances) within the same bound. Our construction of Voronoi diagrams for planar graphs is of ndependent interest. It has already been used to obtain fast exact distance oracles for planar graphs [Cohen-Addad et al., FOCS'17].

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 | 495-514 |

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:†Tel AvivUniversity, Blavatnik School of Computer Science, {haimk,michas}@post.tau.ac.il. Partiallysupported byLen Blavatnikand theBlavatnik Research FundinComputer Science atTel Aviv University, by the IsraeliCenters of Research Excellence(I-CORE)program(center No. 4/11), by theIsrael Science Foundation (grant No. 1841-14), by GIF (grantno. 1161 and 1367), and bythe HermannMinkowski-MINERVACenter for Geometry at Tel AvivUniversity.

Funding Information:

‡InterdisciplinaryCenter Herzliya, EfiAraziSchool ofCom-puter Science, smozes@idc.ac.il. Partiallysupported by the Is-raelScience Foundation (grant No. 794/13 and 592/17). 1TheÕ notation hidespolylogarithmicfactors.

Funding Information:

∗UniversityofHaifa, DepartmentofComputer Science, gawry@mimuw.edu.pl,oren@cs.haifa.ac.il. Partially supported by the IsraelScience Foundation (grant No. 794/13 and 592/17).

Publisher Copyright:

© Copyright 2018 by SIAM.

## ASJC Scopus subject areas

- Software
- Mathematics (all)

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