## Abstract

We consider the problem of sampling from a distribution on graphs, specifically when the distribution is defined by an evolving graph model, and consider the time, space and randomness complexities of such samplers. In the standard approach, the whole graph is chosen randomly according to the randomized evolving process, stored in full, and then queries on the sampled graph are answered by simply accessing the stored graph. This may require prohibitive amounts of time, space and random bits, especially when only a small number of queries are actually issued. Instead, we propose to generate the graph on-the-fly, in response to queries, and therefore to require amounts of time, space, and random bits which are a function of the actual number of queries. We focus on two random graph models: the Barabási-Albert Preferential Attachment model (BA-graphs) [3] and the random recursive tree model [24]. We give on-the-fly generation algorithms for both models. With probability 1 - 1/poly(n), each and every query is answered in polylog(n) time, and the increase in space and the number of random bits consumed by any single query are both polylog(n), where n denotes the number of vertices in the graph. Our results show that, although the BA random graph model is defined by a sequential process, efficient random access to the graph's nodes is possible. In addition to the conceptual contribution, efficient on-the-fly generation of random graphs can serve as a tool for the efficient simulation of sublinear algorithms over large BA-graphs, and the efficient estimation of their performance on such graphs.

Original language | English |
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Title of host publication | 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017 |

Editors | Anca Muscholl, Piotr Indyk, Fabian Kuhn, Ioannis Chatzigiannakis |

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

ISBN (Electronic) | 9783959770415 |

DOIs | |

State | Published - 1 Jul 2017 |

Externally published | Yes |

Event | 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017 - Warsaw, Poland Duration: 10 Jul 2017 → 14 Jul 2017 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 80 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017 |
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Country/Territory | Poland |

City | Warsaw |

Period | 10/07/17 → 14/07/17 |

### Bibliographical note

Publisher Copyright:© Guy Even, Reut Levi, Moti Medina, and Adi Rosén;.

## Keywords

- Local computation algorithms
- Preferential attachment graphs
- Random recursive trees
- Sublinear algorithms