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The Arboricity Captures the Complexity of Sampling Edges

Authors Talya Eden, Dana Ron, Will Rosenbaum

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Streaming, sublinear and near linear time algorithms
  • Theory of computation → Sketching and sampling
Keywords
  • sampling
  • graph algorithms
  • arboricity
  • sublinear-time algorithms

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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).

Cite As Get BibTex

Talya Eden, Dana Ron, and Will Rosenbaum. The Arboricity Captures the Complexity of Sampling Edges. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 52:1-52:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.ICALP.2019.52

Author Details

Talya Eden
  • Tel Aviv University, Tel Aviv, Israel
Dana Ron
  • Tel Aviv University, Tel Aviv, Israel
Will Rosenbaum
  • Max Planck Institute for Informatics, Saarbrücken, Germany

Funding

  • Eden, Talya: Supported by the Azrieli fellowship program for graduate students, by the Sephora Scholarship and by the Weinstein Graduate Studies Prize.
  • Ron, Dana: Partially supported by the Israel Science Foundation grant No. 1146/18.

References

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