## 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.

Original language | English |
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Title of host publication | Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2021 |

Editors | Mary Wootters, Laura Sanita |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959772075 |

DOIs | |

State | Published - 1 Sep 2021 |

Externally published | Yes |

Event | 24th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2021 and 25th International Conference on Randomization and Computation, RANDOM 2021 - Virtual, Seattle, United States Duration: 16 Aug 2021 → 18 Aug 2021 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 207 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 24th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2021 and 25th International Conference on Randomization and Computation, RANDOM 2021 |
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Country/Territory | United States |

City | Virtual, Seattle |

Period | 16/08/21 → 18/08/21 |

### Funding

Funders | Funder number |
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National Science Foundation | CCF-1740751 |

Ben-Gurion University of the Negev | DMS 2022448, CCF-2006664 |

## Keywords

- Graph algorithm
- Sampling edges
- Sublinear algorithms