Tight Analysis for the 3Majority Consensus Dynamics
Abstract
We present a tight analysis for the wellstudied randomized 3majority dynamics of stabilizing consensus, hence answering the main open question of Becchetti et al. [SODA'16]. Consider a distributed system of n nodes, each initially holding an opinion in {1, 2, ..., k}. The system should converge to a setting where all (noncorrupted) nodes hold the same opinion. This consensus opinion should be \emph{valid}, meaning that it should be among the initially supported opinions, and the (fast) convergence should happen even in the presence of a malicious adversary who can corrupt a bounded number of nodes per round and in particular modify their opinions. A wellstudied distributed algorithm for this problem is the 3majority dynamics, which works as follows: per round, each node gathers three opinions  say by taking its own and two of other nodes sampled at random  and then it sets its opinion equal to the majority of this set; ties are broken arbitrarily, e.g., towards the node's own opinion. Becchetti et al. [SODA'16] showed that the 3majority dynamics converges to consensus in O((k^2\sqrt{\log n} + k\log n)(k+\log n)) rounds, even in the presence of a limited adversary. We prove that, even with a stronger adversary, the convergence happens within O(k\log n) rounds. This bound is known to be optimal.
 Publication:

arXiv eprints
 Pub Date:
 May 2017
 arXiv:
 arXiv:1705.05583
 Bibcode:
 2017arXiv170505583G
 Keywords:

 Computer Science  Distributed;
 Parallel;
 and Cluster Computing;
 Computer Science  Discrete Mathematics