License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2020.31
URN: urn:nbn:de:0030-drops-128970
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12897/
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Charalampopoulos, Panagiotis ; Karczmarz, Adam

Single-Source Shortest Paths and Strong Connectivity in Dynamic Planar Graphs

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Abstract

Efficient algorithms for computing and processing additively weighted Voronoi diagrams on planar graphs have been instrumental in obtaining several recent breakthrough results, most notably the almost-optimal exact distance oracle for planar graphs [Charalampopoulos et al., STOC'19], and subquadratic algorithms for planar diameter [Cabello, SODA'17, Gawrychowski et al., SODA'18]. In this paper, we show how Voronoi diagrams can be useful in obtaining dynamic planar graph algorithms and apply them to classical problems such as dynamic single-source shortest paths and dynamic strongly connected components.
First, we give a fully dynamic single-source shortest paths data structure for planar weighted digraphs with Õ(n^{4/5}) worst-case update time and O(log² n) query time. Here, a single update can either change the graph by inserting or deleting an edge, or reset the source s of interest. All known non-trivial planarity-exploiting exact dynamic single-source shortest paths algorithms to date had polynomial query time. Further, note that a data structure with strongly sublinear update time capable of answering distance queries between all pairs of vertices in polylogarithmic time would refute the APSP conjecture [Abboud and Dahlgaard, FOCS'16].
Somewhat surprisingly, the Voronoi diagram based approach we take for single-source shortest paths can also be used in the fully dynamic strongly connected components problem. In particular, we obtain a data structure maintaining a planar digraph under edge insertions and deletions, capable of returning the identifier of the strongly connected component of any query vertex. The worst-case update and query time bounds are the same as for our single-source distance oracle. To the best of our knowledge, this is the first fully dynamic strong-connectivity algorithm achieving both sublinear update time and polylogarithmic query time for an important class of digraphs.

BibTeX - Entry

@InProceedings{charalampopoulos_et_al:LIPIcs:2020:12897,
  author =	{Panagiotis Charalampopoulos and Adam Karczmarz},
  title =	{{Single-Source Shortest Paths and Strong Connectivity in Dynamic Planar Graphs}},
  booktitle =	{28th Annual European Symposium on Algorithms (ESA 2020)},
  pages =	{31:1--31:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-162-7},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{173},
  editor =	{Fabrizio Grandoni and Grzegorz Herman and Peter Sanders},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12897},
  URN =		{urn:nbn:de:0030-drops-128970},
  doi =		{10.4230/LIPIcs.ESA.2020.31},
  annote =	{Keywords: dynamic graph algorithms, planar graphs, single-source shortest paths, strong connectivity}
}

Keywords: dynamic graph algorithms, planar graphs, single-source shortest paths, strong connectivity
Collection: 28th Annual European Symposium on Algorithms (ESA 2020)
Issue Date: 2020
Date of publication: 26.08.2020

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