License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2021.4
URN: urn:nbn:de:0030-drops-145858
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14585/
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Akav, Maor ; Roditty, Liam

A Unified Approach for All Pairs Approximate Shortest Paths in Weighted Undirected Graphs

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LIPIcs-ESA-2021-4.pdf (0.8 MB)

Abstract

Let G = (V,E) be a weighted undirected graph with n vertices and m edges, and let d_G(u,v) be the length of the shortest path between u and v in G. In this paper we present a unified approach for obtaining algorithms for all pairs approximate shortest paths in weighted undirected graphs. For every integer k ≥ 2 we show that there is an Õ(n²+kn^{2-3/k}m^{2/k}) expected running time algorithm that computes a matrix M such that for every u,v ∈ V:

d_G(u,v) ≤ M[u,v] ≤ (2+(k-2)/k)d_G(u,v).

Previous algorithms obtained only specific approximation factors. Baswana and Kavitha [FOCS 2006, SICOMP 2010] presented a 2-approximation algorithm with expected running time of Õ(n²+m√ n) and a 7/3-approximation algorithm with expected running time of Õ(n²+m^{2/3}n). Their results improved upon the results of Cohen and Zwick [SODA 1997, JoA 2001] for graphs with m = o(n²). Kavitha [FSTTCS 2007, Algorithmica 2012] presented a 5/2-approximation algorithm with expected running time of Õ(n^{9/4}).
For k = 2 and k = 3 our result gives the algorithms of Baswana and Kavitha. For k = 4, we get a 5/2-approximation algorithm with Õ(n^{5/4}m^{1/2}) expected running time. This improves upon the running time of Õ(n^{9/4}) due to Kavitha, when m = o(n²).
Our unified approach reveals that all previous algorithms are a part of a family of algorithms that exhibit a smooth tradeoff between approximation of 2 and 3, and are not sporadic unrelated results. Moreover, our new algorithm uses, among other ideas, the celebrated approximate distance oracles of Thorup and Zwick [STOC 2001, JACM 2005] in a non standard way, which we believe is of independent interest, due to their extensive usage in a variety of applications.

BibTeX - Entry

@InProceedings{akav_et_al:LIPIcs.ESA.2021.4,
  author =	{Akav, Maor and Roditty, Liam},
  title =	{{A Unified Approach for All Pairs Approximate Shortest Paths in Weighted Undirected Graphs}},
  booktitle =	{29th Annual European Symposium on Algorithms (ESA 2021)},
  pages =	{4:1--4:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-204-4},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{204},
  editor =	{Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14585},
  URN =		{urn:nbn:de:0030-drops-145858},
  doi =		{10.4230/LIPIcs.ESA.2021.4},
  annote =	{Keywords: Graph algorithms, Approximate All Pairs of Shortest Paths, Distance Oracles}
}

Keywords: Graph algorithms, Approximate All Pairs of Shortest Paths, Distance Oracles
Collection: 29th Annual European Symposium on Algorithms (ESA 2021)
Issue Date: 2021
Date of publication: 31.08.2021

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