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 onthefly, 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 BarabásiAlbert Preferential Attachment model (BAgraphs) and the random recursive tree model. We give onthefly generation algorithms for both models. With probability 11/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 onthefly generation of random graphs can serve as a tool for the efficient simulation of sublinear algorithms over large BAgraphs, and the efficient estimation of their performance on such graphs.
BibTeX  Entry
@InProceedings{even_et_al:LIPIcs:2017:7424,
author = {Guy Even and Reut Levi and Moti Medina and Adi Ros{\'e}n},
title = {{Sublinear Random Access Generators for Preferential Attachment Graphs}},
booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
pages = {6:16:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770415},
ISSN = {18688969},
year = {2017},
volume = {80},
editor = {Ioannis Chatzigiannakis and Piotr Indyk and Fabian Kuhn and Anca Muscholl},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7424},
URN = {urn:nbn:de:0030drops74242},
doi = {10.4230/LIPIcs.ICALP.2017.6},
annote = {Keywords: local computation algorithms, preferential attachment graphs, random recursive trees, sublinear algorithms}
}
Keywords: 

local computation algorithms, preferential attachment graphs, random recursive trees, sublinear algorithms 
Collection: 

44th International Colloquium on Automata, Languages, and Programming (ICALP 2017) 
Issue Date: 

2017 
Date of publication: 

07.07.2017 