## Abstract

We present an explicit and 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(nb^{2}) time 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 Õ(n^{5}/^{3}) time. This improves the recent breakthrough result of Cabello [SODA 2017, SIAM, Philadelphia, 2017, pp. 2143–2152], both by improving the running time (from Õ(n^{11}/^{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 compute, for every vertex v, both the farthest vertex from v (i.e., the eccentricity of v), and the sum of distances from v to all other vertices. Hence, our algorithm can also compute the radius, median, and Wiener index (sum of all pairwise distances) of a planar graph within the same time bounds. Our construction of Voronoi diagrams for planar graphs is of independent interest.

Original language | English |
---|---|

Pages (from-to) | 509-554 |

Number of pages | 46 |

Journal | SIAM Journal on Computing |

Volume | 50 |

Issue number | 2 |

DOIs | |

State | Published - 2021 |

### Bibliographical note

Funding Information:Funding: The second author was partially supported by the Israel Science Foundation (grants 1841/14 and 1595/19), and by GIF (grants 1161 and 1367). The fourth author was partially supported by the Israel Science Foundation (grants 892/13 and 260/18), by GIF (grant 1367), by Len Blavatnik and the Blavatnik Research Fund in Computer Science at Tel Aviv University, by the Israeli Centers of Research Excellence (I-CORE) program (center 4/11), and by the Hermann MinkowskiMINERVA Center for Geometry at Tel Aviv University. The third and fifth authors were partially supported by the Israel Science Foundation (grants 794/13 and 592/17).

Publisher Copyright:

© 2021 Society for Industrial and Applied Mathematics

## Keywords

- Diameter
- Divide-and-conquer
- Planar graph
- Shortest paths
- Voronoi diagrams

## ASJC Scopus subject areas

- General Computer Science
- General Mathematics

## Fingerprint

Dive into the research topics of 'Voronoi diagrams on planar graphs, and computing the diameter in deterministic Õ(n^{5}/

^{3}) time

^{∗}'. Together they form a unique fingerprint.