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 a setting where one generates parts of the sampled graph on-the-fly, in response to queries, and therefore requires amounts of time, space, and random bits that are a function of the actual number of queries. Yet, the responses to the queries correspond to a graph sampled from the distribution in question.Within this framework, we focus on two random graph models: the Barabási-Albert Preferential Attachment model (BA-graphs) (Science, 286 (5439):509-512) (for the special case of out-degree 1) and the random recursive tree model (Theory of Probability and Mathematical Statistics, (51):1-28). 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 work thus proposes a new approach for the access to huge graphs sampled from a given distribution, and 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 on such graphs.
| Original language | English |
|---|---|
| Article number | 28 |
| Journal | ACM Transactions on Algorithms |
| Volume | 17 |
| Issue number | 4 |
| DOIs | |
| State | Published - Oct 2021 |
Bibliographical note
Publisher Copyright:© 2021 Association for Computing Machinery.
Funding
A preliminary version of this work appeared in the Proceedings of ICALP 2017 [11]. Adi Rosén research supported in part by ANR project RDAM. This research was supported by Israel Science Foundation grant Nos. 1867/20 and 867/19. Authors’ addresses: G. Even, Tel Aviv University Tel-Aviv 69978, Israel; email: [email protected]; R. Levi, The Interdisciplinary Center Herzliya (IDC), P.O. Box 167 Herzliya 46150, Israel; email: [email protected]; M. Medina, Faculty of Engineering, Bar-Ilan University, Ramat Gan, 5290002 Israel; email: [email protected]; A. Rosén, CNRS and Université de Paris 75205 Paris Cedex 13, France; email: [email protected]. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]. © 2021 Association for Computing Machinery. 1549-6325/2021/10-ART28 $15.00 https://doi.org/10.1145/3464958
| Funders | Funder number |
|---|---|
| Agence Nationale de la Recherche | |
| Israel Science Foundation | 1867/20, 867/19 |
Keywords
- Random Graph Generator
- local computation algorithms
- preferential attachment graphs
- sublinear algorithms