## Abstract

The Voronoi diagrams technique, introduced by Cabello [SODA’17] to compute the diameter of planar graphs in subquadratic time, has revolutionized the field of distance computations in planar graphs. We present novel applications of this technique in static, fault-tolerant, and partially-dynamic undirected unweighted planar graphs, as well as some new limitations. In the static case, we give n^{3+}o^{(1)}/D^{2} and Õ(n·D^{2}) time algorithms for computing the diameter of a planar graph G with diameter D. These are faster than the state of the art Õ(n^{5}/^{3}) [SODA’18] when D < n^{1}/^{3} or D > n^{2}/^{3}. In the fault-tolerant setting, we give an n^{7}/3+o^{(1)} time algorithm for computing the diameter of G \ {e} for every edge e in G (the replacement diameter problem). This should be compared with the naive Õ(n^{8}/^{3}) time algorithm that runs the static algorithm for every edge. In the incremental setting, where we wish to maintain the diameter while adding edges, we present an algorithm with total running time n^{7}/3+o^{(1)}. This should be compared with the naive Õ(n^{8}/^{3}) time algorithm that runs the static algorithm after every update. We give a lower bound (conditioned on the SETH) ruling out an amortized O(n^{1−ε}) update time for maintaining the diameter in weighted planar graph. The lower bound holds even for incremental or decremental updates. Our upper bounds are obtained by novel uses and manipulations of Voronoi diagrams. These include maintaining the Voronoi diagram when edges of the graph are deleted, allowing the sites of the Voronoi diagram to lie on a BFS tree level (rather than on boundaries of r-division), and a new reduction from incremental diameter to incremental distance oracles that could be of interest beyond planar graphs. Our lower bound is the first lower bound for a dynamic planar graph problem that is conditioned on the SETH.

Original language | English |
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Title of host publication | 31st Annual European Symposium on Algorithms, ESA 2023 |

Editors | Inge Li Gortz, Martin Farach-Colton, Simon J. Puglisi, Grzegorz Herman |

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

ISBN (Electronic) | 9783959772952 |

DOIs | |

State | Published - Sep 2023 |

Event | 31st Annual European Symposium on Algorithms, ESA 2023 - Amsterdam, Netherlands Duration: 4 Sep 2023 → 6 Sep 2023 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 274 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 31st Annual European Symposium on Algorithms, ESA 2023 |
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Country/Territory | Netherlands |

City | Amsterdam |

Period | 4/09/23 → 6/09/23 |

### Bibliographical note

Funding Information:Funding Amir Abboud: Supported by an Alon scholarship and a research grant from the Center for New Scientists at the Weizmann Institute of Science. Shay Mozes: Israel Science Foundation grant 810/21. Oren Weimann: Israel Science Foundation grant 810/21.

Publisher Copyright:

© Amir Abboud, Shay Mozes, and Oren Weimann.

## Keywords

- diameter
- dynamic graphs
- fault tolerance
- Planar graphs

## ASJC Scopus subject areas

- Software