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.ICALP.2020.29
URN: urn:nbn:de:0030-drops-124365
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2020/12436/
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Chechik, Shiri ; Nechushtan, Moran

Simplifying and Unifying Replacement Paths Algorithms in Weighted Directed Graphs

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LIPIcs-ICALP-2020-29.pdf (0.9 MB)

Abstract

In the replacement paths (RP) problem we are given a graph G and a shortest path P between two nodes s and t . The goal is to find for every edge e ∈ P, a shortest path from s to t that avoids e. The first result of this paper is a simple reduction from the RP problem to the problem of computing shortest cycles for all nodes on a shortest path.
Using this simple reduction we unify and extremely simplify two state of the art solutions for two different well-studied variants of the RP problem.
In the first variant (algebraic) we show that by using at most n queries to the Yuster-Zwick distance oracle [FOCS 2005], one can solve the the RP problem for a given directed graph with integer edge weights in the range [-M,M] in Õ(M n^ω) time . This improves the running time of the state of the art algorithm of Vassilevska Williams [SODA 2011] by a factor of log⁶n.
In the second variant (planar) we show that by using the algorithm of Klein for the multiple-source shortest paths problem (MSSP) [SODA 2005] one can solve the RP problem for directed planar graph with non negative edge weights in O (n log n) time. This matches the state of the art algorithm of Wulff-Nilsen [SODA 2010], but with arguably much simpler algorithm and analysis.

BibTeX - Entry

@InProceedings{chechik_et_al:LIPIcs:2020:12436,
  author =	{Shiri Chechik and Moran Nechushtan},
  title =	{{Simplifying and Unifying Replacement Paths Algorithms in Weighted Directed Graphs}},
  booktitle =	{47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)},
  pages =	{29:1--29:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-138-2},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{168},
  editor =	{Artur Czumaj and Anuj Dawar and Emanuela Merelli},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2020/12436},
  URN =		{urn:nbn:de:0030-drops-124365},
  doi =		{10.4230/LIPIcs.ICALP.2020.29},
  annote =	{Keywords: Fault tolerance, Distance oracle, Planar graph}
}

Keywords: Fault tolerance, Distance oracle, Planar graph
Collection: 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)
Issue Date: 2020
Date of publication: 29.06.2020

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