What Else Can Voronoi Diagrams Do for Diameter in Planar Graphs?

Amir Abboud, Shay Mozes, Oren Weimann

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

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 n3+o(1)/D2 and Õ(n·D2) time algorithms for computing the diameter of a planar graph G with diameter D. These are faster than the state of the art Õ(n5/3) [SODA’18] when D < n1/3 or D > n2/3. In the fault-tolerant setting, we give an n7/3+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(n8/3) 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 n7/3+o(1). This should be compared with the naive Õ(n8/3) time algorithm that runs the static algorithm after every update. We give a lower bound (conditioned on the SETH) ruling out an amortized O(n1−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 languageEnglish
Title of host publication31st Annual European Symposium on Algorithms, ESA 2023
EditorsInge Li Gortz, Martin Farach-Colton, Simon J. Puglisi, Grzegorz Herman
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772952
DOIs
StatePublished - Sep 2023
Event31st Annual European Symposium on Algorithms, ESA 2023 - Amsterdam, Netherlands
Duration: 4 Sep 20236 Sep 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume274
ISSN (Print)1868-8969

Conference

Conference31st Annual European Symposium on Algorithms, ESA 2023
Country/TerritoryNetherlands
CityAmsterdam
Period4/09/236/09/23

Bibliographical note

Publisher Copyright:
© Amir Abboud, Shay Mozes, and Oren Weimann.

Keywords

  • Planar graphs
  • diameter
  • dynamic graphs
  • fault tolerance

ASJC Scopus subject areas

  • Software

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