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.2019.52
URN: urn:nbn:de:0030-drops-106287
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10628/
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Eden, Talya ; Ron, Dana ; Rosenbaum, Will

The Arboricity Captures the Complexity of Sampling Edges

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LIPIcs-ICALP-2019-52.pdf (0.5 MB)

Abstract

In this paper, we revisit the problem of sampling edges in an unknown graph G = (V, E) from a distribution that is (pointwise) almost uniform over E. We consider the case where there is some a priori upper bound on the arboriciy of G. Given query access to a graph G over n vertices and of average degree {d} and arboricity at most alpha, we design an algorithm that performs O(alpha/d * {log^3 n}/epsilon) queries in expectation and returns an edge in the graph such that every edge e in E is sampled with probability (1 +/- epsilon)/m. The algorithm performs two types of queries: degree queries and neighbor queries. We show that the upper bound is tight (up to poly-logarithmic factors and the dependence in epsilon), as Omega(alpha/d) queries are necessary for the easier task of sampling edges from any distribution over E that is close to uniform in total variational distance. We also prove that even if G is a tree (i.e., alpha = 1 so that alpha/d = Theta(1)), Omega({log n}/{loglog n}) queries are necessary to sample an edge from any distribution that is pointwise close to uniform, thus establishing that a poly(log n) factor is necessary for constant alpha. Finally we show how our algorithm can be applied to obtain a new result on approximately counting subgraphs, based on the recent work of Assadi, Kapralov, and Khanna (ITCS, 2019).

BibTeX - Entry

@InProceedings{eden_et_al:LIPIcs:2019:10628,
  author =	{Talya Eden and Dana Ron and Will Rosenbaum},
  title =	{{The Arboricity Captures the Complexity of Sampling Edges}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{52:1--52:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10628},
  URN =		{urn:nbn:de:0030-drops-106287},
  doi =		{10.4230/LIPIcs.ICALP.2019.52},
  annote =	{Keywords: sampling, graph algorithms, arboricity, sublinear-time algorithms}
}

Keywords: sampling, graph algorithms, arboricity, sublinear-time algorithms
Collection: 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)
Issue Date: 2019
Date of publication: 04.07.2019

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