### 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 language | English (US) |
---|---|

Title of host publication | Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 |

Pages | 541-550 |

Number of pages | 10 |

State | Published - 2012 |

Event | 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 - Kyoto, Japan Duration: Jan 17 2012 → Jan 19 2012 |

### Other

Other | 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012 |
---|---|

Country | Japan |

City | Kyoto |

Period | 1/17/12 → 1/19/12 |

### Fingerprint

### ASJC Scopus subject areas

- Software
- Mathematics(all)

### Cite this

*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.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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.

}

TY - GEN

T1 - Stochastic coalescence in logarithmic time

AU - Loh, Po Shen

AU - Lubetzky, Eyal

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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.

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UR - http://www.scopus.com/inward/citedby.url?scp=84860148933&partnerID=8YFLogxK

M3 - Conference contribution

AN - SCOPUS:84860148933

SN - 9781611972108

SP - 541

EP - 550

BT - Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012

ER -