Stochastic coalescence in logarithmic time

11 years 6 months ago

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The following distributed coalescence protocol was introduced by Dahlia Malkhi in 2006 motivated by applications in social networking. Initially there are n agents wishing to coalesce into one cluster via a decentralized stochastic process, where each round is as follows: Every cluster ﬂips a fair coin to dictate whether it is to issue or accept requests in this round. Issuing a request amounts to contacting a cluster randomly chosen proportionally to its size. A cluster accepting requests is to select an incoming one uniformly (if there are such) and merge with that cluster. Empirical results by Fernandess and Malkhi suggested the protocol concludes in O(log n) rounds with high probability, whereas numerical estimates by Oded Schramm, based on an ingenious analytic approximation, suggested that the coalescence time should be super-logarithmic. Our contribution is a rigorous study of the stochastic coalescence process with two consequences. First, we conﬁrm that the above process ...

Po-Shen Loh, Eyal Lubetzky

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