TY - JOUR

T1 - Sublinear Random Access Generators for Preferential Attachment Graphs

AU - Even, Guy

AU - Levi, Reut

AU - Medina, Moti

AU - Rosén, Adi

N1 - Publisher Copyright:
© 2021 Association for Computing Machinery.

PY - 2021/10

Y1 - 2021/10

N2 - 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.

AB - 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.

KW - Random Graph Generator

KW - local computation algorithms

KW - preferential attachment graphs

KW - sublinear algorithms

UR - http://www.scopus.com/inward/record.url?scp=85117077673&partnerID=8YFLogxK

U2 - 10.1145/3464958

DO - 10.1145/3464958

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AN - SCOPUS:85117077673

SN - 1549-6325

VL - 17

JO - ACM Transactions on Algorithms

JF - ACM Transactions on Algorithms

IS - 4

M1 - 28

ER -