Data structures and algorithms for disjoint set union problems
ACM Computing Surveys (CSUR)
Probabilistic analysis of disjoint set union algorithms
SIAM Journal on Computing
Fast perfection-information leader-election protocol with linear immunity
STOC '93 Proceedings of the twenty-fifth annual ACM symposium on Theory of computing
Simple and efficient leader election in the full information model
STOC '94 Proceedings of the twenty-sixth annual ACM symposium on Theory of computing
Randomness-optimal sampling, extractors, and constructive leader election
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Perfect information leader election in log * n+0(1) rounds
Journal of Computer and System Sciences
On the average behavior of set merging algorithms (Extended Abstract)
STOC '76 Proceedings of the eighth annual ACM symposium on Theory of computing
On spreading recommendations via social gossip
Proceedings of the twentieth annual symposium on Parallelism in algorithms and architectures
Parallel tree contraction and its application
SFCS '85 Proceedings of the 26th Annual Symposium on Foundations of Computer Science
An optimal randomized parallel algorithm for finding connected components in a graph
SFCS '86 Proceedings of the 27th Annual Symposium on Foundations of Computer Science
Combinatorica
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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 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.