## 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 α, we design an algorithm that performs O ^{α}_{d} · ^{log}_{ε}^{3 n} queries in expectation and returns an edge in the graph such that every edge e ∈ E is sampled with probability (1 ± ε)/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 ε), as Ω ^{α}_{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., α = 1 so that ^{α}_{d} = Θ(1)), Ω _{loglog}^{log n}_{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 α. 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).

Original language | English |
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Title of host publication | 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019 |

Editors | Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, Stefano Leonardi |

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

ISBN (Electronic) | 9783959771092 |

DOIs | |

State | Published - 1 Jul 2019 |

Externally published | Yes |

Event | 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019 - Patras, Greece Duration: 9 Jul 2019 → 12 Jul 2019 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 132 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019 |
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Country/Territory | Greece |

City | Patras |

Period | 9/07/19 → 12/07/19 |

### Bibliographical note

Publisher Copyright:© Graham Cormode, Jacques Dark, and Christian Konrad; licensed under Creative Commons License CC-BY

## Keywords

- Arboricity
- Graph algorithms
- Sampling
- Sublinear-time algorithms