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.
Highlights
Consider a Markov process in which a sequence {St}t of states, St ∈ S, evolves over time t ≥ 1
We focus on two random graph models: the Barabási-Albert Preferential Attachment model (BA-graphs) [3] and the random recursive tree model [24]
The BA random graph model is defined by a sequential process, efficient random access to the graph’s nodes is possible
Summary
Consider a Markov process in which a sequence {St}t of states, St ∈ S, evolves over time t ≥ 1. We are interested in the following resources of a graph generator: (1) the number of random bits consumed per query, (2) the running time per query, and (3) the increase in memory space per query. Our main result is a randomized on-the-fly graph generator for BA-graphs over n vertices that answers adjacency list queries. Vertices randomly select their “children” among the “future” vertices, while maintaining the same probability distribution as if each child picked “in the future” its parent We apply these techniques in the related model of random recursive trees [24] ( used within the evolving copying model [10]), and use them as a building block for our main result for BA-graphs. Support of adjacency list queries is especially useful for simulating (partial) DFS and BFS over graphs
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