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Sampling Multiple Edges Efficiently

Authors Talya Eden , Saleet Mossel, Ronitt Rubinfeld

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ACM Subject Classification
  • Theory of computation → Sketching and sampling
Keywords
  • Sampling edges
  • graph algorithm
  • sublinear algorithms

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Abstract

We present a sublinear time algorithm that allows one to sample multiple edges from a distribution that is pointwise ε-close to the uniform distribution, in an amortized-efficient fashion. We consider the adjacency list query model, where access to a graph G is given via degree and neighbor queries.
The problem of sampling a single edge in this model has been raised by Eden and Rosenbaum (SOSA 18). Let n and m denote the number of vertices and edges of G, respectively. Eden and Rosenbaum provided upper and lower bounds of Θ^*(n/√ m) for sampling a single edge in general graphs (where O^*(⋅) suppresses poly(1/ε) and poly(log n) dependencies). We ask whether the query complexity lower bound for sampling a single edge can be circumvented when multiple samples are required. That is, can we get an improved amortized per-sample cost if we allow a preprocessing phase? We answer in the affirmative. 
We present an algorithm that, if one knows the number of required samples q in advance, has an overall cost that is sublinear in q, namely, O^*(√ q ⋅(n/√ m)), which is strictly preferable to O^*(q⋅ (n/√ m)) cost resulting from q invocations of the algorithm by Eden and Rosenbaum. 
Subsequent to a preliminary version of this work, Tětek and Thorup (arXiv, preprint) proved that this bound is essentially optimal.

Cite As Get BibTex

Talya Eden, Saleet Mossel, and Ronitt Rubinfeld. Sampling Multiple Edges Efficiently. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 51:1-51:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.51

Author Details

Talya Eden
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Saleet Mossel
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA
Ronitt Rubinfeld
  • CSAIL, Massachusetts Institute of Technology, Cambridge, MA, USA

Funding

  • Eden, Talya: This work was supported by the NSF Grant CCF-1740751, the Eric and Wendy Schmidt Fund, and Ben-Gurion University.
  • Rubinfeld, Ronitt: This work was supported by the NSF TRIPODS program (awards CCF-1740751 and DMS 2022448), NSF award CCF-2006664 and by the Fintech@CSAIL Initiative.

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