Stochastic coalescence in logarithmic time

Po Shen Loh, Eyal Lubetzky

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

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 indeed requires super-logarithmic time w.h.p., where the inefficient rounds are due to oversized clusters that occasionally develop. Second, we remedy this by showing that a simple modification produces an essentially optimal distributed protocol: If clusters favor their smallest incoming merge request then the process does terminate in O(log n) rounds w.h.p., and simulations show that the new protocol readily outperforms the original one. Our upper bound hinges on a potential function involving the logarithm of the number of clusters and the cluster-susceptibility, carefully chosen to form a supermartingale. The analysis of the lower bound builds upon the novel approach of Schramm which may find additional applications: Rather than seeking a single parameter that controls the system behavior, instead one approximates the system by the Laplace transform of the entire cluster-size distribution.

Original languageEnglish (US)
Title of host publicationProceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012
Pages541-550
Number of pages10
StatePublished - 2012
Event23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 - Kyoto, Japan
Duration: Jan 17 2012Jan 19 2012

Other

Other23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012
CountryJapan
CityKyoto
Period1/17/121/19/12

Fingerprint

Coalescence
Logarithmic
Laplace transforms
Hinges
Random processes
Supermartingale
Distributed Protocol
Social Networking
Number of Clusters
Terminate
Flip
Potential Function
Logarithm
Laplace transform
Control Parameter
Susceptibility
Decentralized
Stochastic Processes
Entire
Lower bound

ASJC Scopus subject areas

  • Software
  • Mathematics(all)

Cite this

Loh, P. S., & Lubetzky, E. (2012). Stochastic coalescence in logarithmic time. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 (pp. 541-550)

Stochastic coalescence in logarithmic time. / Loh, Po Shen; Lubetzky, Eyal.

Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012. 2012. p. 541-550.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Loh, PS & Lubetzky, E 2012, Stochastic coalescence in logarithmic time. in Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012. pp. 541-550, 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, Kyoto, Japan, 1/17/12.
Loh PS, Lubetzky E. Stochastic coalescence in logarithmic time. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012. 2012. p. 541-550
Loh, Po Shen ; Lubetzky, Eyal. / Stochastic coalescence in logarithmic time. Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012. 2012. pp. 541-550
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